A Gain Scheduling Approach to The Load Frequency Control ...

A Gain Scheduling Approach to The LoadFrequency Control in Smart Grids

by

Shichao Liu, B. Eng., M. Eng. (Research)

A Thesis submitted to

the Faculty of Graduate and Postdoctoral Affairs

in partial fulfillment of

the requirements for the degree of

Doctor of Philosophy

in

Electrical and Computer Engineering

Ottawa-Carleton Institute for Electrical and Computer Engineering (OCIECE)

Department of Systems and Computer Engineering

Carleton University

September 2014

Copyright c©2014 - Shichao Liu

The undersigned recommend to

the Faculty of Graduate and Postdoctoral Affairs

acceptance of the Thesis

A Gain Scheduling Approach to The Load FrequencyControl in Smart Grids

Submitted by Shichao Liu

in partial fulfilment of the requirements for the degree of

Doctor of Philosophy

Professor Peter X. Liu, Supervisor

Professor A. El Saddik, Supervisor

Ottawa-Carleton Institute for Electrical and Computer Engineering (OCIECE)

Department of Systems and Computer Engineering

Carleton University

2014

ii

Abstract

As an indispensable technology that enables smart grid operations, the

two-way communication networking technology greatly facilitates the vast amount

of information exchange involved in the operations. However, this technology also

makes it challenging to ensure the reliability and stability of smart grids. It is

well-known that communication networks, especially wireless networks, are unreliable

because of communication delays and random communication failures. If these

factors are not properly considered in the control scheme of a smart grid, they may

degrade the dynamic performance of a power system and/or even make the entire

power system unstable.

In this thesis, two important issues related to the effects of these unreliable

network-associated factors on the load frequency control of smart grids are inves-

tigated. One of them is that how communication delays can affect the load frequency

control of low-voltage microgrids. To study this issue, a thorough small-signal analy-

sis is presented for an islanded microgrid. By conducting this analysis, the maximal

communication delay below which the microgrid can maintain stable (usually de-

fined as a delay margin) is determined and its relationships with secondary frequency

control gains are identified. To improve the robustness of the microgrid to commu-

nication delays, a gain scheduling method is proposed for the load frequency control.

Simulation results of the Canadian urban benchmark distribution system verify the

correctness of the small-signal analysis results and the effectiveness of the proposed

gain scheduling load frequency control. The other is that how communication failures

can influence the load frequency control of high-voltage largely interconnected power

grids. To investigate this issue, two particular scenarios that can result in communi-

cation failures are considered, including cognitive radio networks and denial of service

attacks. By modeling power systems with communication failures as linear switched

iii

systems, the effects of these two scenarios on the load frequency control of largely

interconnected power systems are respectively analyzed. To compensate the effects

of the communication failures, a distributed gain scheduling method is also proposed

for the load frequency control. Simulation results of a four-area interconnected power

system show the proposed gain scheduling control can greatly improve its robustness

to communication failures.

iv

Acknowledgments

I would like to extend my great thanks and sincere gratitude to my co-supervisors,

Professor Peter X. Liu and Professor Abdulmotaleb El Saddik, for their excellent

guidance. Without their strong and persistent support and encouragement, I would

never have completed this work.

Many thanks are due to the committee members: Dr. Chunsheng Yang from

National Research Council Canada, Dr. Shevin Shirmohammadi from the Depart-

ment of Electrical Engineering and Computer Science at the University of Ottawa,

Dr. Xiaoyu Wang from the Department of Electronics at Carleton University, and

Dr. Jerome Talim from the Department of Systems and Computer Engineering at

Carleton University. Their constructive comments are very helpful in improving the

presentation of this thesis.

I thank Dr. Minyi Huang from the School of Mathematics and Statistics at Car-

leton University for his inspiring discussions on stochastic games and dynamic pro-

gramming.

I also thank my friends and colleagues in my office: Dr. Shafiqul Islam, Dr. Jason

Paul Rhinelander, Mr. Kun Wang, for their great help when I was carrying this

research.

The financial supports of Carleton University President 2010 PhD Fellowship and

Department Scholarship are gratefully acknowledged.

Finally, I would like to thank my family and relatives for their support.

v

Table of Contents

Abstract iii

Acknowledgments v

Table of Contents vi

List of Tables xi

List of Figures xii

List of Acronyms xvi

List of Symbols xviii

1 Introduction 1

1.1 Smart Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Smart Grid Technology . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 Smart Grid Communication Requirements and Architecture . 4

1.2 Load Frequency Control in Smart Grids . . . . . . . . . . . . . . . . 6

1.2.1 Load Frequency Control in High-voltage Largely Interconnected

Power Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.2 Load Frequency Control in Microgrids . . . . . . . . . . . . . 8

vi

1.3 Literature Review: Challenges for Smart Grid Control over Open Com-

munication Links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.1 Communication Delays in Power Grids . . . . . . . . . . . . . 11

1.3.2 Communication Failures in Power Grids . . . . . . . . . . . . 12

1.3.3 Control Algorithms for Compensating The Two Communica-

tion Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 The Effect of Communication Delays on Load Frequency Control

in An Islanded Microgrid 16

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 The Studied Microgrid System . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Small-Signal Model of The Microgrid . . . . . . . . . . . . . . . . . . 18

2.3.1 Model of The Inverter-based DG with Two-level Controllers . 19

2.3.2 Network Model . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.3 Interface Equations . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.4 Small-signal Analysis of The Load Frequency Control . . . . . 23

2.4 The Effect of Time Delays on The Microgrid Stability . . . . . . . . 27

2.4.1 Determination of Delay Margin . . . . . . . . . . . . . . . . . 27

2.4.2 Relationships between Load Frequency Control Gains and De-

lay Margins . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5 Validation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 Gain Scheduling Approach for Compensating The Communication

Delay Effect on Load Frequency Control of An Islanded Microgrid 37

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

vii

3.2 General Control Structure of An Islanded Microgrid with PMUs and

Gain Schedulers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Gain Scheduling Methodology . . . . . . . . . . . . . . . . . . . . . . 39

3.3.1 Feasible Gain Sets . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3.2 Feasible Gains with Respect to The Microgrid Performance . . 43

3.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Stability Analysis of Load Frequency Control over Cognitive Radio

Networks in Largely Interconnected Power Grids 51

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2 Cognitive Radio Networks in Smart Grids . . . . . . . . . . . . . . . 52

4.3 Modeling of The LFC over A Cognitive Radio Network in A Largely

Interconnected Power System . . . . . . . . . . . . . . . . . . . . . . 54

4.3.1 The Model of Cognitive Radio Networks . . . . . . . . . . . . 55

4.3.2 The Switched System Model for The LFC over A Cognitive

Radio Network . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.4 Stabilities of LFC over Cognitive Radio Networks . . . . . . . . . . . 62

4.4.1 Asymptotical Stability for Arbitrary but Bounded Sojourn Times 62

4.4.2 Mean-square Stability for Random Sojourn Times with Inde-

pendent Identical Distribution . . . . . . . . . . . . . . . . . . 64

4.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5 Denial-of-Service Attacks on Load Frequency Control in Largely

Interconnected Power Grids 80

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2 Denial-of-Service (DOS) Attacks in Smart Grids . . . . . . . . . . . . 81

viii

5.3 Modeling of A Largely Interconnected Power System with DoS Attacks 82

5.4 Existence of Successful DoS Attacks in the Smart Grid . . . . . . . . 85

5.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6 Modeling and Distributed Gain Scheduling Strategy for Load Fre-

quency Control with Communication Failures in Largely Intercon-

nected Power Grids 91

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.2 General Structure of The Proposed Distributed Gain Scheduling Strategy 93

6.3 Modeling of A Multi-area Interconnected Power System with Commu-

nication Failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.4 Stability Analysis of The Multi-area Interconnected Power System with

Communication Failures . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.5 Distributed Gain Scheduling Strategy for The LFC in A Smart Grid . 100

6.6 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7 Conclusions 111

7.1 Thesis Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.2 Possible Directions for Future Research . . . . . . . . . . . . . . . . . 113

A Microgrid Parameters 115

B Chebyshev’s Differentiation Matrix 116

C Two-area Power System Parameters 118

D Four-area power system parameters 119

ix

E Publications 120

List of References 121

x

List of Tables

4.1 The maximum eigenvalues of the matrices λ(Ψ(τ)) with different sam-

pling periods Ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2 The maximum singular values σmax(Ψ(τ)) with the maximum sojourn

times τmax under sampling period Ts . . . . . . . . . . . . . . . . . . 78

4.3 The maximum singular values σmax(Υ(τ)) with the maximum sojourn

times τmax under sampling period Ts . . . . . . . . . . . . . . . . . . 78

A.1 Distribution system parameters . . . . . . . . . . . . . . . . . . . . . 115

A.2 Inverter parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

C.1 Two-area power system parameters . . . . . . . . . . . . . . . . . . . 118

D.1 Four-area power system parameters . . . . . . . . . . . . . . . . . . . 119

xi

List of Figures

1.1 The structure of a smart grid . . . . . . . . . . . . . . . . . . . . . . 3

1.2 The main characteristics of a smart grid compared with a traditional

power grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Three-layers of a load frequency control system . . . . . . . . . . . . 7

1.4 Two-level hierarchical load frequency control of a microgrid . . . . . . 9

2.1 The Canadian urban benchmark distribution system . . . . . . . . . 17

2.2 Structure of the multi-DG system model . . . . . . . . . . . . . . . . 18

2.3 Structure of the multi-DG system two-level control . . . . . . . . . . 19

2.4 Reference frame transformation . . . . . . . . . . . . . . . . . . . . . 23

2.5 Root loci of the multi-DG system with Kiω = 60 . . . . . . . . . . . . 25

2.6 Root loci of the critical eigenvalues with Kiω = 60 . . . . . . . . . . . 25

2.7 Root loci of the multi-DG system with Kpω = 2 . . . . . . . . . . . . 26

2.8 Root loci of the critical eigenvalues with Kpω = 2 . . . . . . . . . . . 26

2.9 Root loci of Δ(η) when the secondary frequency control gains are

Kpω = 2 and Kiω = 60 . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.10 Relationship between delay margin τd and secondary frequency control

gains Kpω and Kiω . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.11 Structure of the simulation platform in Matlab/SimPower . . . . . . 31

2.12 Dynamic performance of the microgrid when τ = 0s . . . . . . . . . . 33

2.13 Dynamic performance of the microgrid when τ = 0.1s . . . . . . . . . 33

xii

2.14 Dynamic performance of the microgrid when τ = 0.15s . . . . . . . . 34

2.15 Dynamic performance of the microgrid when τ = 0.21s . . . . . . . . 34

2.16 Dynamic performance of the microgrid in Case 5 . . . . . . . . . . . . 35

2.17 Dynamic performance of the microgrid in Case 6 . . . . . . . . . . . . 36

3.1 Structure of the multi-DG system model . . . . . . . . . . . . . . . . 38

3.2 Root locus for βiω under different time delays (Arrows direct the in-

creasing gains) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3 Root locus for βpω under different time delays (Arrows direct the in-

creasing gains) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.4 Cost curve with respect to βiω1 when Kpω = 2 . . . . . . . . . . . . . 44

3.5 Cost curve with respect to βpω1 when Kiω = 60 . . . . . . . . . . . . . 44

3.6 Dynamic performance of DG1 with τ = 0.1s . . . . . . . . . . . . . . 46

3.7 Dynamic performance of DG1 with τ = 0.2s . . . . . . . . . . . . . . 46

3.8 The dynamic of the time-varying delay (only showing first 100 samples) 48

3.9 Dynamic performances of DG1 with a βiω1 gain-scheduler . . . . . . 48

3.10 The dynamic of the βiω1 gain-scheduler (only showing first 100 samples) 49

3.11 Dynamic performances of DG1 with a βpω1 gain-scheduler . . . . . . 49

3.12 The dynamic of the βpω1 gain-scheduler (only showing first 100 samples) 50

4.1 Two-area power system over cognitive radio (CR) networks . . . . . . 55

4.2 The cognitive radio channel illustration . . . . . . . . . . . . . . . . . 56

4.3 The proposed On-Off cognitive channel model . . . . . . . . . . . . . 56

4.4 The block diagram of the control area i . . . . . . . . . . . . . . . . . 59

4.5 Two-area power system . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.6 Mean square errors of the power system states with uniform p.d.f so-

journ times in Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 72



xiii





4.10 Mean square errors of the power system states with geometric p.d.f

sojourn times in Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 74







5.1 Two-area load frequency control (LFC) under DoS attacks . . . . . . 82

5.2 The model of the power system under DoS attacks . . . . . . . . . . 84

5.3 The root locus of the average two-area power system (Arrows indicate

α decreasing) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.4 The dynamics of area 1 under different DoS attacks initial times . . . 89

5.5 The dynamics of area 2 under different DoS attacks initial times . . . 90

6.1 Centralized control scheme . . . . . . . . . . . . . . . . . . . . . . . . 92

6.2 Distributed control scheme . . . . . . . . . . . . . . . . . . . . . . . . 92

6.3 The proposed distributed control algorithm . . . . . . . . . . . . . . 93

6.4 Six communication topologies . . . . . . . . . . . . . . . . . . . . . . 106

6.5 The scheduling scheme of the 5 imperfect communication topology modes106

6.6 Dynamic response of Area 1 . . . . . . . . . . . . . . . . . . . . . . . 107




xiv

6.10 Dynamic responses of 4 areas under the distributed gain scheduling

strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

xv

List of Acronyms

Acronyms Definition

AGC automatic generation control

ACC area control center

AMI advanced metering infrastructure

CR cognitive radio

CTD communication topology detector

DG distributed generation

DS distributed storage

EMS energy management system

HAN home area network

IED intelligent electronic devices

LFC load frequency control

LC local controller

LTI linear time-invariant

xvi

LQR Linear quadratic regulators

MGCC microgrid centralized controller

MC Markov chain

MSE mean square error

NAN neighborhood area network

NCS networked control system

PMU phasor measurement units

PV photovoltaic

PLL phase-locked loop

PU primary user

QoS quality of service

RTU remote terminal unit

SCADA supervisory control and data acquisition

SU secondary users

WAN wide area network

WAMCP wide-area monitoring, control and protection

WAMCS wide area monitor and control systems

ZOH zero-order-holds

xvii

List of Symbols

Symbols Definition

A state matrix of descriptor system small-signal models

Aτ state matrix of delayed system small-signal models

B input matrix of descriptor system small-signal models

C0, S0 diagonal matrices of dq-to-xy transformation

C, C, and M matrices for estimating eigenvalues of small signal models

DN Chebyshev’s differentiation matrix of dimension {N +1}×{N + 1}

Di equivalent damping coefficient

E singular matrix of descriptor system small-signal models

ei frequency errors

Gi, Hi symmetric positive definite matrices

idrefi, iqrefi inverter current regulator current set point

idi, iqi inverter output currents on dq-axis

xviii

ix, iy network currents on x − y frame

J cost function

Kpii, Kiii inverter current controller gains

Kpωi, Kiωi secondary frequency controller gains

KpP LLi and KiP LLi PLL controller gains

Kiωi, Kpωi equivalent gains of the secondary frequency controller with

gain schedulers

K∗ij optimal inter-connective gains

L(k) communication matrix

lij(k) elements in the communication matrix

P DCrefi inverter corrective power set point

P SFrefi inverter supplementary real power set point

P DCrefi corrective real power set point

Pi, Qi instantaneous real and reactive power

Qrefi reactive power set point

Ri speed droop coefficient

S1, S2 switch postions

Tchitime constant of turbine

Tij synchronizing power coefficient

xix

tk, tk+1 two consecutive state jump instants

Tgitime constant of governor

u(l) the state feedback controller at time slot l ∈ [tk, tk+1)

vdi, vqi inverter terminal voltages on dq-axis

V (x(k)) composite Lyapunov function of the linear system

Vi(xi(k)) Lyapunov function of each subsystem in the linear system

Vx, Vy network voltages on x − y frame

x small-signal variable vector

x(t − τ) small-signal variable vector

x(k) state vector of the system under DoS attacks

z(k) augmented state

ω0 nominal frequency reference

ωi instantaneous frequency

ωP LLi inverter terminal voltage frequency acquired by PLL

Δ small-signal variables

δi inverter terminal voltage phase angle on x − y frame

τ time delay

λ system eigenvalue

τd delay margin

xx

τci critical time delays

βiωi, βpωi gain scheduling variables

ω0i frequency of the original DG

ωdi frequency of the original DG with communication delays

ΔPmigenerator mechanical power deviation

ΔPviturbine valve position deviation

Δfi frequency deviation

ΔPciangular velocity

ΔP itie net tie-line power flow

ΔPLiload deviation

θk communication channel state at the kth time slot

{τ1, τ2, · · · , τi, · · · , τk+1} sojourn time sequence

τi sojourn time varible

τmin, τmax sojourn time bounds

σi switch position variable

Φ1, Φ2 system matrices with augmented state

α normalized DoS attacks launching time

σi switch position variable

λm(∗), λM(∗) the minimum and maximum eigenvalues of matrix ∗

xxi

Chapter 1

Introduction

1.1 Smart Grids

As the next generation of power grids, smart grids have been attracting increasing

attention from both academic and industrial communities around the world. In this

chapter, technologies that enable smart grid operations are summarized. Among

these technologies, the two-way communication technology plays a very important

role in building smart grids. The requirements and architectures of the smart grid

communication are also introduced. As one critical smart grid operation, load fre-

quency control (LFC) in both large interconnected power systems and microgrids

is discussed. Challenges brought by the two-way communication technology in the

power system control are also presented. Finally, the thesis structure is summarized.

1.1.1 Smart Grid Technology

Most of power systems around the world have been in existence for many decades

since they were developed. With the rapid development of industries and the living

conditions of human beings, the need for energy has grown tremendously and the

operational scenarios are quite different from which they were. Nowadays, traditional

1

CHAPTER 1. INTRODUCTION 2

power systems are facing several unique challenges. According to [1, 2], these chal-

lenges include:

(1) The deregulation of the power industry, which divides the utility company mo-

nopolies by separating the production of energy from its distribution. This results in

large uncertainties of power flow scenarios in the power industry.

(2) The increasingly large penetration of renewable energy, such as wind and solar

energy, to achieve sustainable growth and minimize environmental impact. This fur-

ther increases the uncertainty in power supply.

(3) The increasing demand for a highly reliable and efficient power supply, to support

both industrial and everyday power utilization.

(4) The threat of possible attacks on either the physical or the cyber assets of the

power grid.

Apparently, the traditional power systems are infeasible to handle these chal-

lenges. This can be seen from several large blackouts which happened recently, such

as the 2003 North American, 2003 European and 2012 Indian blackouts [3,4]. There

is thus a quite urgent demand for new and effective solutions to the monitoring and

operation of large-scale power systems. Upgrading the traditional power systems into

smart grids is increasingly recognized by industry and many national governments

as the answer to address these challenges. According to the definition given by the

U.S. Department of Energy (DOE) Office of Electricity Delivery and Energy Relia-

bility, a smart grid integrates advanced two-way communication network technology

and intelligent computer processing technology into the current power systems, from

large-scale generation through delivery systems to electricity consumers [5], shown in

Fig 1.1. A comparison between the main characteristics of a traditional power grid

and a smart grid is made in Fig 1.2.


Figure 1.1: The structure of a smart grid

Figure 1.2: The main characteristics of a smart grid compared with a traditionalpower grid


1.1.2 Smart Grid Communication Requirements and Archi-

tecture

In this section, critical requirements for communications in smart grids are listed and

a three-layer smart grid communication architecture is described in detail.

Requirements:

Several important requirements that the smart grid communication have to meet are

listed as follows.

Security : Advanced two-way communication and intelligent computation tech-

nologies are critical to enable smart grid applications and functionalities, such as

wide-area monitoring, control and protection (WAMCP), distributed generation man-

agement, advanced metering infrastructure (AMI), real-time pricing, etc. While these

technologies facilitate the aggregation and communication of both system-wide infor-

mation and local measurement data in selected locations, they expose smart grids

to both cyber attacks and physical attacks. There were several reported attacks on

power grids in U.S. [6, 7]. The compromise of communication networks and/or com-

puters will severely endanger the stability and reliability of smart grid monitoring and

control functionalities. Therefore, developing efficient security mechanism for smart

grid communication networks and computers is one of the most critical issues.

Quality of Service (QoS) : QoS in communication networks refers to latency,

bit-error rate, packet-loss rate, throughput, jitter, connect outage probability etc.. In

smart grids, different applications have different QoS specifications. Some applica-

tions are time sensitive, such as inter-area oscillation damping control in wide-area

monitoring, control and protection (WAMCP) and anti-islanding protection in dis-

tributed generation management. Low latency should be carefully ensured in these

applications. Other applications could be quality sensitive, such as dynamic stability


assessment in the energy management system (EMS). Specifying the QoS require-

ments for communication networks according to a variety of smart grid applications,

is important and challenging. In order to fulfill this task, both simulations of power

dynamics and field tests of demo projects are necessary.

Scalability : Smart grid applications involve a vast amount of devices such as

smart meters, intelligent sensors, data collectors, electric vehicles, and distributed

generators. The amount of these devices will keep increasing. Therefore, a smart grid

should be able to handle the scalability issue as more and more network nodes will

be integrated into the smart grid.

Inter-operability : Smart grid communication networks need to be able to

support the data flow over the whole smart grid, from bulk generations through de-

livery units (transmission systems and distribution systems) to electricity consumers.

A three-layer hybrid communication architecture which include wide area network

(WAN), neighborhood area network (NAN) and home area network (HAN) should

be used to support this largely geographical coverage of data flow. This hybrid com-

munication architecture has to involve a large variety of communication standards

in order to be implemented in practice. The inter-operability among these various

communication standards and subnetworks should be guaranteed, in order to keep

smart grid monitoring and control stable and reliable. Many regulation groups, such

as GridWise architecture council and NIST, are working collaboratively to address

the inter-operability issue. NIST even announced an IEEE P2030 inter-operability

project in June 2009.

Smart Grid Communication Architecture

The three-layer communication architecture for smart grids includes wide area net-

work (WAN), neighborhood area network (NAN) and home area network (HAN).

Wide area network (WAN) : The upper layer of the three-layer communication


architecture is WAN. It provides communication networks for upstream utility assets

such as power plants, distributed generation sources, distributed storage units, sub-

stations and so on. The communication standards that can be used for WAN include

fiber optics, WiMAX, power line communication (PLC), satellite communication and

cellular communications.

Neighbor area network (NAN) : The middle layer of the three-layer commu-

nication architecture is NAN. It supplies communication networks for smart meters,

field components, and gateways that form the backbone of the network between dis-

tribution system substations and HANs. The smart grid standards for NAN include

WiMax, PLC, and Ethernet.

Home area network (HAN) : The lower layer of the smart grid communication

architecture is HAN. It creates communications among home appliances including

sensors, monitors, loads, etc.. The candidates of networking standards for HAN

include ZigBee, WiFi, HomePlug, etc.

1.2 Load Frequency Control in Smart Grids

The frequency and power generation control in a power system is usually referred

to load frequency control (LFC). It mainly keeps the frequency of the power system

at a nominal value (i.e. 60Hz) by adjusting power generation set point. In this

section, LFC in both high-voltage largely interconnected power grids and low-voltage

microgrids is described.

1.2.1 Load Frequency Control in High-voltage Largely Inter-

connected Power Grids

The LFC is the major function of automatic generation control (AGC) systems in

largely inter-connected power grids. It is also fundamental in determining the way in


Figure 1.3: Three-layers of a load frequency control system

which the frequency will change when load changes happen. The main objectives of

the LFC in largely inter-connected power grids are summarized as follows [8]:

(1) Maintain frequency at the scheduled point;

(2) Maintain the net tie-line power interchanges with neighboring control areas at

their scheduled values;

(3) Maintain power allocation among generators in accordance with area dispatching

needs. The structure of the LFC is illustrated in Fig 1.3. As shown in this figure,

there are following three control layers in the LFC.

Primary Control: It is the turbine governing system, which is decentralized because

it is installed in power plants situated at different geographical areas, The action of

turbine governors due to frequency changes when reference values of regulators are

kept constant is referred as primary frequency control;

Secondary Control: It is composed of frequency control and tie-line control to force

primary control to eliminate the frequency and net tie-line interchange deviations.


Being equipped only with primary controllers, the power system will not be able to

return to the initial frequency without any additional action when a change in total

demand happens. In order to eliminate the frequency and net tie-line interchange

deviations, an area control error is defined as ACE = ΔPT L −βΔf . ACE is combined

of frequency bias Δf and net tie-line power deviations ΔPT L, while β is a weighting

parameter. By zeroing the ACE, the objective of eliminating the frequency and net

tie-line power biases can be fulfilled;

Tertiary Control: Tertiary control sets the reference values of power in individual

generating units to the values calculated by optimal dispatch in such a way that

the overall demand is satisfied together with the schedule of power interchanges. By

adjusting manually or autonomously the set points of individual turbine governors,

tertiary control ensures the following:

(1) Adequate spinning reserve in the units participating in primary control;

(2) Optimal dispatch of units participating in secondary control;

(3) Restoration of the bandwidth of secondary control in a given cycle.

In the LFC, there are two data exchange loops. One of them is the feed-forward

loop in which control centers send control signals to remote terminal units (RTUs)

and turbine governors of local power plants. The other is the feedback loop where

measurement signals are transmitted from RTUs to the control centers over commu-

nication links. The open communication links used in these two loops can facilitate

data exchange in the LFC.

1.2.2 Load Frequency Control in Microgrids

Being different with the high-voltage interconnected power systems mentioned in

the previous section, a microgrid is usually used to better organize the low-voltage

distribution system. It is an integrated energy system comprising interconnected

loads, distributed generation (DG) and distributed storage (DS) units [2,9,10]. There


Figure 1.4: Two-level hierarchical load frequency control of a microgrid

are three operation modes for a microgrid, including the grid-connected mode, the

islanded mode and the transition mode between the former two modes [11]. In the

grid-connected mode, the frequency of the microgrid is synchronized with the nominal

frequency of the main grid. The microgrid only adjusts real and reactive power

profiles by injecting a certain amount of real and reactive power into the main grid

or absorbing from it. However, in the islanded mode, due to lack of the main grid as

a reference, excursions of the frequency of the microgrid may happen if there is no

proper frequency control in the microgrid [12].

Being similar to the frequency control of high-voltage interconnected power sys-

tems, the LFC of a microgrid also has a hierarchical structure shown in Fig 1.4. The

two-level hierarchical control structure includes

Primary Control: A local controller (LC) for each inverter-based distributed gen-

erator usually acts fast to counteract an imbalance between load and generation. The

LC usually refers to an inverter controller due to the fact that a DG here is interfaced

with the prime mover via a power-electronic inverter, including a constant power

controller and a droop controller, as shown in Fig 1.4. However, since there is no


reference frequency for the microgrid to follow when it is in islanded mode, with only

LCs, the microgrid frequency is not able to return to the given nominal reference.

With the help of a secondary frequency controller, the frequency of the microgrid

can be restored to its nominal frequency. It will facilitate the reconnection of the

microgrid to the main grid when the reconnection is detected to be applicable.

Secondary Control: A microgrid centralized controller (MGCC) is located at the

low voltage side of a substation in a microgrid and only runs when the microgrid is

in the islanded mode. To restore the frequency of the microgrid to the nominal value

fixed by the utility, the MGCC generates supplementary real power set points for LCs

of DGs and DSs and sends them to the corresponding LCs through low bandwidth

communication channels.

1.3 Literature Review: Challenges for Smart Grid

Control over Open Communication Links

In microgrids, the LFC involves communications between MGCC and LCs. In in-

terconnected power grids, the LFC also needs the communication between control

centers and RTUs. When open communication infrastructures are embedded into

these power grids, they will support the vast amount of data exchange. However,

there are several inherently unreliable factors existing in these open communication

links, mainly including communication delays and communication failures. It is crit-

ical to understand the effects of the two factors on the power system control. In

this section, existing work in the literature on the smart grid control with these two

factors are summarized.


1.3.1 Communication Delays in Power Grids

When the effect of the communication delay on largely interconnected power systems

is considered, it mainly refers to its effect on wide-area monitor and control systems

(WAMCS). A communication delay in the WAMCS consists of two parts, including

one communication delay caused when control centers send control signals to RTUs

and the other caused when measurements signals are transmitted from RTUs to the

control centers.

The existence of time delays in communication links may endanger the stability of

power systems and degrade their dynamic performance. The impact of time delays on

the wide-area control design of power systems is firstly investigated by H. Wu and G.T.

Heydt [13]. In this work, it is shown that the overshoot value of active power variation

increases and the settling time is lengthened in the presence of time delays which are

modeled by Pade approximations. Instead of using the simple Pade approximation, a

more complicated stochastic model is proposed to study the effect of communication

delays on wide-area damping control by J. W. Stahlhut and G.T. Heydt et. al [14].

In their work, it is found that the time delays of control signals can degrade the

performance of a wide area control system, by modeling the time delay as a M/M/1

queue. For the LFC in largely interconnected power systems, G/G/1 queues are used

to model both constant and random time delays with different network protocols [15].

In practice, communication delays are investigated and tested in China Southern

Grid [16]. In their tests, the Ethernet is used as the communication network. The

measured time delay varies from 60 ms to 210 ms. Although the previous results are

very promising, they have been obtained only by time domain simulations and field

tests. Solid theoretical analysis of the time delay effect on power system was still

missing until recently. In 2012, the impact of time delays on power system stability

is analyzed via small-signal models by F. Milano and M. Anghel [17].


The presence of communication delays can harm not only the stability of largely

interconnected power systems, but also low-voltage microgrids. Due to their low iner-

tias, DGs in microgrids can response to disturbances very fast. Therefore, the effect

of communication delays could be more critical than largely interconnected power

grids which usually have large inertias. Recently, it has been found that communica-

tion delays can badly affect load sharing control in microgrids [18, 19]. The effect of

communication delays on load frequency control has been investigated in [20, 21]. It

is noted that communication delays may cause the instability of power systems and

the degradation of power system performances.

1.3.2 Communication Failures in Power Grids

Besides time delays existing in communication links, the stability of power system

control can also be jeopardized by communication failures, especially largely inter-

connected power grids. The communication element failure has been assessed as one

of the dominant risks that could threat the reliability and safety of power system-

s [22, 23]. Series of probabilistic assessment studies reveal that the increase of the

communication failure rate results in an increase of the load shedding when system

operators try to restore the stability of a power system [24]. Besides the probabilistic

methods, by conducting time-domain simulations of power systems, it is also found

that communication failures can disturb power system state estimations and cause

load losses [25,26]. Cascade failures in power systems could even be worsen if commu-

nication failures happen during the restoration process [27]. In [28], communication

infrastructures are investigated in the IEEE 118-bus test network for both centralized

and decentralized control strategies. It emphasizes that the communication failures of

a power grid may cause very serious problems for both system operation and control.


1.3.3 Control Algorithms for Compensating The Two Com-

munication Factors

Since communication delays and failures may create stability issues in power systems,

increasing research efforts have been devoted to designing advanced control methods

to overcome the communication-induced instability. For the compensation of commu-

nication delays, some of these control methods are robust control in which time delays

are dealt as uncertainties to power systems, such as sliding mode control [29], H2/H∞

control [30, 31], and delay-dependant control [32, 33]. Other control methods include

smith predictor [34], hierarchical control [35, 36], model predictive control [37], and

Kalman filter [38]. For counteracting the effect of communication failures, there are

also a few advanced control methods in existence. For instance, a joint control and

communication topology design is formulated as mixed-integer optimization problem

for distributed damping control of power systems [39]. For a wide-area power system,

a redundant supplementary damping controller is designed to achieve the resiliency

to communication failures [40]. For multiple generators in a distribution system, co-

operative control theory is applied for self-organization of distributed photovoltaic

(PV) generators [41].

While these control algorithms mentioned above are very promising, the commu-

nication delays between centralized controllers and power plants are either estimated

or viewed as uncertainties by the centralized controllers. However, this kind of com-

munication delays can only be measured after control outputs are received in power

plants. Thus, the centralized control outputs cannot exactly compensate this kind of

communication delays. In the thesis, a novel gain scheduling approach is proposed for

each local power plant controller to compensate the communication delays between

a centralized secondary controller and local power plant. Instead of estimating the

communication delay resulted from control output transmission, it can be measured


by marking time stamps when the centralized secondary controller sends control out-

puts and the local controllers embedded with gain schedulers receive them. These

gain schedulers can then adjust the delayed control outputs to compensate the time

delay. A distributed gain scheduling method is also proposed to counteract the effect

of communication failures on the performance of the LFC of largely interconnected

power systems.

1.4 Contributions

While all the work in the literature has made great contributions to understanding

the effects of unreliable factors in open communication links on power system control,

there are at least two issues that have not been well addressed. On the one hand, the

effect of communication delays on the low-voltage microgrid control (in specific, load

frequency control) has only been studied by trial simulations, while a comprehensive

theoretical analysis has still been missing. On the other hand, although a lot of work

on the impact of communication delays on high-voltage largely interconnected power

system control have been done, the communication failure effect on the power system

control, particularly load frequency control, has not been covered.

In the thesis, a set of results are developed that deal with the above two issues

involved in the LFC of smart grids. Several contributions in this thesis are summarized

as follows.

• 1. For the first time, at our best knowledge, a thoroughly theoretical small-signal

analysis of the effect of communication delays on the load frequency control of

an islanded microgrid is presented.

• 2. A gain scheduling method is proposed to improve the robustness of the

microgrid load frequency control to communication delays.


• 3. For the first time, at the best knowledge of authors, the impact of communica-

tion failures on the load frequency control of high-voltage largely interconnected

power grids is considered and comprehensively analyzed.

• 4. A distributed gain scheduling method is developed for the load frequency

control to compensate the impact of communication failures.

1.5 Thesis Organization

The remaining chapters of this thesis are summarized as follows. In Chapter 2, the

effect of communication delays on the LFC in an islanded multi-DG microgrid is

studied. In Chapter 3, based on the small-signal model of the microgrid formulated

in the previous chapter, a gain scheduling approach is proposed to compensate the

communication delay effect on the LFC performance of the microgrid. In Chapter 4,

for largely interconnected power systems, a CR network is considered as the source

of communication failures. By modeling the CR network as a On-Off switch with

sojourn times, a novel switched power system model is proposed for the LFC of

the interconnected power systems. In Chapter 5, the DoS attack is considered as

another reason that results in communication failures for the largely interconnected

power systems. In Chapter 6, a distributed gain scheduling strategy is proposed

to compensate the potential degradation of the performance of the LFC caused by

communication failures in largely interconnected power systems. Finally, in Chapter

7, conclusions of the thesis are summarized.

Chapter 2

The Effect of Communication Delays on

Load Frequency Control in An Islanded

Microgrid

2.1 Introduction

In this chapter, we study the communication delay effect on the stability of the load

frequency control (LFC) in an islanded microgrid. To achieve this objective, a time-

delay small-signal system model is formulated for the microgrid system. Based on the

analysis of this model, the relationships between load frequency control parameters

and delay margins below which the system can stay stable are found. Simulation

studies illustrate the effect of communication delays on the microgrid stability and

validate the proposed small-signal analysis results.

The rest of this chapter is organized as follows. The studied microgrid system

is introduced in Section 2.2. A small-signal model is proposed and the effect of

load frequency control parameters are analyzed in Section 2.3. Delay margins are

determined and the effect of load frequency control parameters on delay margins are

also analyzed in Section 2.4. The analysis results are verified by simulation studies

16

CHAPTER 2. EFFECT OF COMMUNICATION DELAYS ON LFC 17

Figure 2.1: The Canadian urban benchmark distribution system

of a multi-DG microgrid in Section 2.5. Finally, Conclusions are summarized in

Section 2.6.

2.2 The Studied Microgrid System

The Canadian urban benchmark distribution system introduced in [42,43] is used to

investigate the dynamic performance of the LFC here. The schematic diagram of the

test system is shown in Fig 2.1. The data of the system can be seen in Table A.1 and

Table A.2 in Appendix A. The utility source is 120 kV and the 12.5 kV substation is

connected to the grid through a circuit breaker (CB) and a substation transformer

with a capacity of 10 MVA. A 2.75 Mvar capacitor bank is located at the substation.

Four inverter-based DGs (three-phase 208 V) are evenly distributed along the feeder.

They are connected to the feeder through the step-down transformers and the DG

terminals are from Node 6 to Node 9 in sequence. The constant impedance load

model with power factor 0.95 is adopted to represent the local load of each generator.

A microgrid central controller (MGCC) is installed at low-voltage side of the


Figure 2.2: Structure of the multi-DG system model

120kV/12.5kV substation (Bus 1 in Fig 2.1), to manage the operation of the microgird.

The MGCC provides all kinds of references (such as real and reactive power references)

to the local controller (LC) of each DG unit, while each LC sends its measurements

(such as frequency and power signals) back to the MGCC through communication

channels. Among all kinds of functionalities of the MGCC, its load frequency control

is mainly investigated in this work. Although the frequency-droop controller is able to

stabilize the frequency dynamics in case of small disturbances, it cannot remove the

frequency steady-state error to the nominal frequency given by the utility source. In

order to restore the frequency of the microgrid to its nominal set point, a centralized

secondary frequency controller is designed at the MGCC.

2.3 Small-Signal Model of The Microgrid

In this section, the small-signal model of the studied microgrid is presented. Although

it is for the Canadian urban benchmark distribution system, the process of building

this small-signal model is generic and can be extended to any other microgrid. The

model of the microgrid with multiple DGs is shown in Fig 2.2, where n is the number

of DGs and m is the number of loads. The model consists of three blocks, including


Figure 2.3: Structure of the multi-DG system two-level control

the DG block, the network and load block, as well as the interface block.

2.3.1 Model of The Inverter-based DG with Two-level Con-

trollers

The control structure of the microgrid including both MGCC and a LC is shown in

Fig 2.3. The power control loop of the local controller of an DG inverter consists

of a power control and an inner current loop, to regulate the inverter output power

by tracking given real power set points. Both the power and current controller are

Proportional-Integral (PI) controllers.

The power controller is

idrefi = (Kppi +Kipi

s)(P SF

refi + P DCrefi − Pi) (2.1)

iqrefi = (Kppi + Kipi

s)(Qrefi − Qi) (2.2)


where, idrefi and iqrefi are the current set points of the ith DG, Kppi and Kipi are

proportional and integral control gains of the ith DG, respectively, P SFrefi is the ith DG

supplementary real power set point assigned by the secondary frequency controller of

MGCC, P DCrefi is the corrective real power set point generated by the power control of

ith DG, Pi and Qi are the instantaneous real and reactive power. Qrefi is the reactive

power set point of ith DG. Since the focus of this work is the frequency control, only

the real power control structure is shown in Fig 2.3.

The inverter current controller is

vdi = (Kpii + Kiii

s)(idrefi − idi) (2.3)

vqi = (Kpii + Kiii

s)(iqrefi − iqi) (2.4)

where, vdi and vqi are inverter terminal voltages on dq-axis, idi and iqi are the ith

DG inverter output currents on dq-axis, Kpii and Kiii are the gains of the current

controller.

The ω − P characteristic of the frequency droop control can be described as

P DCrefi = Kωi(ω0 − ωi) (2.5)

where, Kωi is the droop control gain, P DCrefi is the corrective power set point due to

frequency variations.

The secondary frequency control is

P SFrefi = (Kpωi + Kiωi

s)(ω0 − ωi) (2.6)

where, P SFrefi is the supplementary power set point of the ith DG assigned by the

secondary frequency controller, Kpωi and Kiωi are proportional and integral control


gains, respectively, ω0 is the nominal frequency reference, ωi is the instantaneous

frequency obtained from a phase-locked loop (PLL).

The PLL model can be expressed as

ωP LLi = (KpP LLi +KiP LLi

s)Vqi − ω0 (2.7)

where, ωP LLi is the inverter terminal voltage frequency acquired by PLL, Vqi is the

q axis voltage obtained by using the abc − dq transformation, ω0 is the nominal

frequency, KpP LLi and KiP LLi are the PI controller gains of the ith DG.

Consider now a microgrid consisting of n+1 buses. The first n buses are connected

with DGs, while the Bus n+1 is the infinite bus (here is the utility generator). Define

variables as

δ = [δ1, δ2, · · · , δn]T , ω = [ω1, ω2, · · · , ωn]T , vd = [vd1, vd2, · · · , vdn]T ,

vq = [vq1, vq2, · · · , vqn]T , Vd = [Vd1, Vd2, · · · , Vdn]T , Vq = [Vq1, Vq2, · · · , Vqn]T ,

id = [id1, id2, · · · , idn]T , iq = [iq1, iq2, · · · , iqn]T , idref = [idref1, idref2, · · · , idrefn]T ,

iqref = [iqref1, iqref2, · · · , iqrefn]T , P = [P1, P2, · · · , Pn]T , Q = [Q1, Q2, · · · , Qn]T ,

Pref = [Pref1, Pref2, · · · , Prefn]T ,

After being linearized around a steady-state operating point, the small-signal e-

quations of the inverter-based DGs are written as

Δδ = Δω (2.8)

Δω − KpP LLΔVq = KiP LLΔVq (2.9)

KpiΔid + Δvd − KpiΔidref = KiiΔidref − KiiΔid (2.10)

KpiΔiq + Δvq − KpiΔiqref = KiiΔiqref − KiiΔiq (2.11)

LsΔid = Δvd (2.12)


LsΔiq = Δvq (2.13)

Δidref + KppΔP = −KipΔP + KipΔPref (2.14)

Δiqref + KppΔQ = −KipΔQ (2.15)

ΔPdref + (Kpω + Kω)Δω = −KiωΔω + KiωΔω0 (2.16)

0 = Vd0Δid + id0ΔVd + Vq0Δiq + iq0ΔVq − ΔP (2.17)

0 = Vd0Δiq + id0ΔVd − Vq0Δid − Id0ΔVq − ΔQ (2.18)

2.3.2 Network Model

The network model in a common reference frame can be written as:

⎡⎢⎢⎢⎢⎣

Δix

Δiy

⎤⎥⎥⎥⎥⎦ =

⎡⎢⎢⎢⎢⎣

G −B

B G

⎤⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎣

ΔVx

ΔVy

⎤⎥⎥⎥⎥⎦ (2.19)

where, the matrices G and B are acquired from the network system admittance matrix

and Vx = [Vx1, Vx2, · · · , Vxn]T , Vy = [Vy1, Vy2, · · · , Vyn]T , ix = [ix1, ix2, · · · , ixn]T ,

iy = [iy1, iy2, · · · , iyn]T .

2.3.3 Interface Equations

Among mathematic equations obtained above, each DG model is developed in its own

d − q reference frame. To develop the small-signal model of the microgrid, all the

voltages and currents must be transformed to the common reference x−y frame. The

reference frame transformation between local d − q reference frame and the common

x − y reference frame is shown in Fig 2.4.


Figure 2.4: Reference frame transformation

The interface equations are

ΔVd = C0ΔVx − Vx0S0Δδ + S0ΔVy + Vy0C0Δδ (2.20)

ΔVq = S0ΔVx − Vx0C0Δδ + C0ΔVy + Vy0S0Δδ (2.21)

Δix = C0Δid − id0S0Δδ − S0Δiq − iq0C0Δδ (2.22)

Δiy = S0Δid + id0C0Δδ + C0Δiq − iq0S0Δδ (2.23)

where δi is the individual inverter terminal voltage phase angle in x − y reference

frame, and the diagonal matrices C0 and S0 are defined as C0 = diag{cos(δi0)} and

S0 = diag{sin(δi0)}, respectively.

2.3.4 Small-signal Analysis of The Load Frequency Control

The state space of the overall system is formulated as the following descriptor system

EΔx = AΔx + Fr0 (2.24)

where x = [δ, ω, id, iq, idref , iqref , vd, vq, P, Q, Pref , Vd, Vq, ix, iy, Vx, Vy]T , r0 =

[ω0]T , E is a parameter matrix which is singular, A is the system matrix, F is a


parameter matrix.

Definition. 2.3.1 [44, 45] The descriptor system (2.24) is asymptotically stable

if all general roots of det(λE − A) = 0 are in the open left-hand plane.

With the small-signal model of the microgrid in hand, we are able to investigate

the effect of the load frequency control gains Kpω and Kiω on the stability of the

microgrid. The Canadian urban benchmark distribution system shown in Fig 2.1 is

considered and the initial power generation reference is P0 = 0.3. Without loss of

generality, four DGs in this microgrid are considered to be identical inverter-based

DGs. Parameters of the system including the distribution system and inverters can

be seen in Table A.1 and Table A.2 in Appendix A.

Firstly, the effect of the proportional gain Kpω on the system stability is analyzed.

The integral gain Kiω is fixed at Kiω = 60. The root loci of det(λE − A) = 0 with

increasing Kpω in the range of [0.1, 2] is shown in Fig 2.5. By analyzing this root loci,

λ11 and λ12 are identified as the critical eigenvalues which are the closest eigenvalues

to the imaginary axis. The root loci of these two eigenvalues with increasing Kpω in

the range of [0.1, 3] is shown in Fig 2.6. It can be seen that λ11 and λ12 are moving

toward the right-half plane as the Kpω increases. The observation of their root loci

shows that the upper bound of Kpω that guarantees the microgrid stable is Kpω = 2.8.

Therefore, the stable proportional gain set is obtained.

Then, the impact of the integral gain Kiω on the system stability is studied. The

proportional gain Kpω is fixed at Kpω = 2, the root loci of det(λE − A) = 0 with

increasing Kiω in the range of [1, 60] is shown in Fig 2.7. The root loci of the critical

eigenvalues λ11 and λ12 with increasing Kiω in the range of [1, 70] is shown in Fig 2.8.

It can been found that λ11 is moving towards the right-half plane as the Kiω increases.

This observation shows the upper bound of Kiω that guarantees the microgrid stable

is Kiω = 66. Thus, the stable integral gain set is also obtained.


−200 −150 −100 −50 0−300

−200

−100

0

100

200

300

Real(1/s)

Imag

(rad

/s)

Figure 2.5: Root loci of the multi-DG system with Kiω = 60

−6 −5 −4 −3 −2 −1 0 1 2−8

−6

−4

−2

0

2

4

6

8

Real(1/s)

Imag

(rad

/s)

Kpω

=2.8

Figure 2.6: Root loci of the critical eigenvalues with Kiω = 60


−150 −100 −50 0−300

−200

−100

0

100

200

300

Real(1/s)

Imag

(rad

/s)

Figure 2.7: Root loci of the multi-DG system with Kpω = 2

−10 −5 0 5−8

−6

−4

−2

0

2

4

6

8

Real(1/s)

Imag

(rad

/s)

Kiω

=66

Figure 2.8: Root loci of the critical eigenvalues with Kpω = 2


2.4 The Effect of Time Delays on The Microgrid

Stability

A total time delay τ is considered to exist in the communication channels between

MGCC and LCs. The overall system model becomes a delayed descriptor system,

written as

EΔx = AΔx + AτΔx(t − τ) + Fr0 (2.25)

where τ denotes a time delay, x(t − τ) is the time delayed state, and Aτ is the state

matrix of the delayed descriptor system.

The characteristic equation of the delayed descriptor system is

det(λE − Δ(λ, τ)) = 0 (2.26)

where

Δ(λ, τ) = A + Aτe−λτ (2.27)

Definition. 2.4.1 [44, 45] For a given τ , the delayed descriptor system (2.25) is

asymptotically stable if all general roots of its characteristic equation (2.26) are in

the open left-hand plane.

Definition. 2.4.2 A critical delay denoted by τd is called a delay margin if the

delayed descriptor system (2.25) is stable for τ < τd and it is unstable for τ > τd.

2.4.1 Determination of Delay Margin

The approaches to determine the delay margin for power systems have been discussed

in [33, 46]. In this work, an eigenvalue approach in [46] is extended to the delayed

descriptor system (2.25).

Given a pair of conjugate eigenvalues on the imaginary axis, they are denoted by


λimag = ±jω. The following equation is satisfied

jω = eig(Δ(ω, τ)) (2.28)

where eig(�) denotes eigenvalues of �, and

Δ(ω, τ) = A + Aτe−jωτ (2.29)

Define a variable η = ωτ . Then, the equation (2.29) can be expressed as

Δ(η) = A + Aτ e−jη (2.30)

e−jη changes periodically with η and the period is 2π. Thus, Δ(η) also changes

periodically with a 2π period. We can change η within one period [0, 2π] and get

the root loci of the eigenvalues of Δ(η) within the range η ∈ [0, 2π]. If there exist

eigenvalues ±jωc on the imaginary axis at ηc, the corresponding critical time delay

τc can be obtained by the following:

τc = ηc/ωc (2.31)

Consider a case there exist communication time delays in the studied system with

the secondary frequency control gains Kpω = 2, Kiω = 60 and initial power reference

P0 = 0.3. For this case, the root loci of the eigenvalues of Δ(η) within the range

η ∈ [0, 2π] is shown in Fig 2.9.

It can be seen from Fig 2.9 that there are two pairs of conjugate eigenvalues on

the imaginary axis denoted as ±jωd1 and ±jωd2. Their corresponding critical time

delays are denoted as τc1 and τc2, respectively. The delay margin τd is the minimum

one between τc1 and τc2, described as τd = min{τc1, τc2}. In this case, τd = 0.2053s.


−30 −20 −10 0 10 20 30−200

−150

−100

−50

0

50

100

150

200

Real(1/s)

Imag

(rad

/s)

jωd2

jωd1

−jωd1

−jωd2

Figure 2.9: Root loci of Δ(η) when the secondary frequency control gains are Kpω = 2and Kiω = 60

Provided there exist L critical time delays denoted as τc1, τc2, · · · , τcL, the delay

margin τd is

τd = min{τc1, τc2, · · · , τcL} (2.32)

2.4.2 Relationships between Load Frequency Control Gains

and Delay Margins

In this subsection, delay margins with respect to different Kpω and Kiω are obtained

for the studied system by using the described method above. The results are shown

in Fig 2.10.

It can be found that the delay margin τd increases with the increase of the pro-

portional gains Kpω when Kiω are fixed, while for fixed Kpω, the delay margin τd

increases with the decrease of the integral gains Kiω. By obtaining the relationships

between the delay margin and load frequency control gains, we can choose proper


0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

Proportion Gain Kpw

Del

ay M

argi

n τ d (s

econ

ds)

Integral Gain Kiw

=60

Integral Gain Kiw

=50

Integral Gain Kiw

=40

Figure 2.10: Relationship between delay margin τd and secondary frequency controlgains Kpω and Kiω

gains for a corresponding delay margin.

In order to guarantee a larger delay margin for the microgrid system, the load

frequency controller should have relatively bigger Kpω and smaller Kiω within their

applicable ranges.

2.5 Validation Studies

The microgrid shown in Fig 2.1 is used to study the effect of communication delays

on dynamic performances of the microgrid with a load frequency controller. Also,

delay margins calculated in the previous section are verified by the corresponding

time domain simulations. The circuit breaker at 120 kV and the 12.5 kV substation

is considered to be initially open that means the microgrid is islanded at t = 0s. The

simulation platform for this microgrid is developed in the Matlab/SimPower R2007b


Figure 2.11: Structure of the simulation platform in Matlab/SimPower

environment shown in Fig 2.11. The load frequency control gains are initially set to

be Kpω = 2 and Kiω = 60 based on the results of the small-signal analysis of the

microgrid. The nominal frequency is chosen as ω0 = 377rad/s (60Hz) for the load

frequency control. This nominal frequency is identical to the current Northern Amer-

ican standard frequency, which guarantees the easy reconnection of the microgrid to

the utility main grid.

Firstly, in order to illustrate the effect of communication delays and verify the

calculated delay margin, the following four cases are investigated.

• Case 1: there is no communication delay in the microgrid;

• Case 2: there is a constant total communication delay τ = 0.1s in the microgrid;

• Case 3: there is a constant total communication delay τ = 0.15s in the micro-

grid;


• Case 4: there is a constant total communication delay τ = 0.21s in the micro-

grid.

The simulation results for these four cases are shown from Fig 2.12 to Fig 2.15,

respectively.

When there is no communication delay in the microgrid, it can be seen that the

frequency dynamic of the microgrid with four identical DGs converge fast, shown in

Fig. 2.12. The steady-state voltages of the four DGs can be kept as the nominal 1pu.

The real power set points for the four DGS are also same.

When the time delay increases to τ = 0.1s and τ = 0.15s, the dynamics of the

microgrid can still converge although they spend longer to damp oscillations, shown

in Fig. 2.13 and Fig. 2.14. It can be found the real power set points for the four DGs

are not identical any more in Fig. 2.13(b) and Fig. 2.14(b). This is the reason that

the nominal frequency set point is the only input to the secondary frequency control,

while real power set points are modified to keep the frequency at the nominal set

point ω0 = 377rad/s. Thus, the function of the secondary frequency controller in the

microgrid is verified.

When we continue increasing the time delay to τ = 0.21s, the dynamic perfor-

mances of the microgrid become unstable, shown in Fig. 2.15. It can be noticed

that the calculated delay margin for Kpω = 2 and Kiω = 60 is τd = 0.2053s, shown

in Fig. 2.10. This calculated delay margin closely coincides with the delay margin

estimated by the time-domain simulation. Therefore, the presented delay margin

calculation method also works well.

In the above cases, the communication delays for all the DGs are considered to

be identical. To evaluate more general cases, the following two cases in which the

communication delay for each DG is different from each other, defining four time

delays as τ1 for DG1, τ2 for DG2, τ3 for DG3, τ4 for DG4, respectively.


0 1 2 3 4 5370375380385

ω (r

ad/s

)

(a) Frequencies of DGs

0 1 2 3 4 50.25

0.3

0.35P

(pu)

(b) Real powers of DGs

0 1 2 3 4 50.9

1

1.1

Time (s)

V (p

u)

(c) Voltages of DGs

DG1−DG4

DG1−DG4

DG1−DG4

Figure 2.12: Dynamic performance of the microgrid when τ = 0s

0 1 2 3 4 5370375380385

ω (r

ad/s

)


0 1 2 3 4 50.2

0.4

P (p

u)


0 1 2 3 4 50.9

1

1.1

Time (s)

V (p

u)

(c) Voltages of DGs

DG1−DG4

DG1−DG4

DG4DG1 DG3 DG2

Figure 2.13: Dynamic performance of the microgrid when τ = 0.1s


0 1 2 3 4 5370375380385

ω (r

ad/s

)


0 1 2 3 4 50.2

0.4

0.6P

(pu)


0 1 2 3 4 50.9

1

1.1

Time (s)

V (p

u)

(c) Voltages of DGs

DG1−DG4

DG1−DG4

DG3DG1DG4 DG2


0 1 2 3 4 5370375380385

ω (r

ad/s

)


0 1 2 3 4 5

0.5

1

P (p

u)


0 1 2 3 4 50.9

1

1.1

Time (s)

V (p

u)

(c) Voltages of DGs

DG1−DG4

DG1−DG4

DG4 DG1 DG3 DG2



0 1 2 3 4 5370375380385

ω (r

ad/s

)


0 1 2 3 4 5

0.40.60.8

P (p

u)


0 1 2 3 4 50.9

1

1.1

Time (s)

V (p

u)

(c) Voltages of DGs

DG3 DG2

DG1−DG4

DG1−DG4

DG4DG1

Figure 2.16: Dynamic performance of the microgrid in Case 5

• Case 5: four time delays are less than the delay margin, τ1 = 0.1s, τ2 = 0.15s,

τ3 = 0.1s, τ4 = 0.18s ;

• Case 6: the time delay for DG4 is larger than the delay margin τ4 = 0.25s,

while τ1 = 0.1s, τ2 = 0.15s, τ3 = 0.1s.

The results are shown in Fig. 2.16 and Fig. 2.17. It can be seen from Fig. 2.16 that

the dynamics of the microgrid are still stable when the four different time delays are

shorter than the delay margin. However, the observations from Fig. 2.17 show that

the microgrid becomes unstable as long as the communication delay for one DG is

longer than the delay margin (such as DG4 in Case 6).

2.6 Summary

The impact of communication delays on an islanded multi-DG microgrid with a load

frequency control is studied in this chapter. Based on a small-signal model of the


0 1 2 3 4 5370375380385

ω (r

ad/s

)


0 1 2 3 4 5

0.5

1

P (p

u)


0 1 2 3 4 50.9

1

1.1

Time (s)

V (p

u)

(c) Voltages of DGs

DG3

DG1−DG4

DG4

DG1−DG4

DG2DG1

Figure 2.17: Dynamic performance of the microgrid in Case 6

microgrid without considering communication delays, the effect of the load frequency

control gains on the microgrid stability is firstly analyzed. A delayed small-signal

system model is then formulated for this microgrid system with communication de-

lays. By tracing critical eigenvalues of the characteristic equation of this model, a

delay margin which indicates the maximum communication delay that the microgrid

maintains stable is determined. For Kpω = 2 and Kiω = 60, the maximum allowable

communication delay for the microgrid is τd = 0.2053s. By conducting an extensive

sensitivity study, it has been found that the delay margin increases with the increase

of the proportional gains while it decreases with the increase of the integral gains.

A validation study of a microgrid with 4 inverter-based DGs is also conducted. It

has illustrated the effect of communication delays on the stability of the microgrid

load frequency control and the effectiveness of the obtained method to determine the

delay margin. It has also verified the relationships between the delay margin and the

load frequency control gains.

Chapter 3

Gain Scheduling Approach for

Compensating The Communication Delay

Effect on Load Frequency Control of An

Islanded Microgrid

3.1 Introduction

In the previous chapter, it has been found that communication delays can badly

affect the dynamic performance of an islanded microgrid. Therefore, it is critical

to develop advanced control algorithms for the LFC of the microgrid to compensate

the communication delay effect. To handle this issue, a gain scheduling approach

is proposed in this chapter. Studies of an islanded microgrid with 4 inverter-based

DGs show the proposed gain scheduling method can greatly improve the dynamic

performance of the microgrid, compared to a fixed gain load frequency controller.

The rest of this chapter is organized as follows. In Section 3.2, the general control

structure is described. In Section 3.3, the proposed gain scheduling method is pre-

sented. In Section 3.4, simulations of an islanded microgrid are used to evaluate the

effectiveness of the proposed gain scheduling method. Finally, conclusions are made

37

CHAPTER 3. GAIN SCHEDULING APPROACH FOR LFC 38

Figure 3.1: Structure of the multi-DG system model

in Section 3.5.

3.2 General Control Structure of An Islanded Mi-

crogrid with PMUs and Gain Schedulers

A gain scheduling approach is proposed to compensate the effect of communication

delays on the dynamic performance of the islanded microgrid shown in Fig 2.1. The

microgrid control structure including both the MGCC and one LC is shown in Fig 3.1.

The secondary frequency controller is a proportion and integral (PI) controller. This

PI controller adjusts the real power set points for each DG to restore their frequencies

to the nominal one and sends them to each LC. For each LC, it is equipped with a

phasor measurement unit (PMU) and a gain scheduler embedded with a GPS receiver.

When PMUs measure frequencies at each DG bus of the microgrid, they mark these


measurements with time stamps generated by GPS. After the MGCC receives these

frequency measurements and calculates real power set points for each LC, it sends

these set points to LCs. Since the time spent in MGCC calculation is very short,

it is omitted when we consider time delays in the microgrid. In each LC, a gain

scheduler receives the corresponding real power set point and marks it also with a

time stamp. By comparing the time stamps marked by PMUs and by gain schedulers,

round-trip communication delays are calculated. Then, gain schedulers in LCs adjust

corresponding gains and generate new real power set points to compensate these

communication delays. The gain adjustment is done according to feasible gain sets

which are obtained by offline root locus analysis and trial simulations, with a quadratic

state error as the cost index.

3.3 Gain Scheduling Methodology

Due to the presence of communication delays in the load frequency control loop, the

performance of the original microgrid may be degraded. In order to remain good

system performances, with respect to a certain cost function, the load frequency

controller gains in the microgrid need to be adjusted according to the measured com-

munication delays. In this section, we add a local gain scheduler denoted by a variable

βωi in each DG controller of the microgrid, to compensate the degradation of the mi-

crogrid performance. The integral gain scheduling variable βiωi and proportional gain

scheduling variable βpωi are investigated, respectively.

3.3.1 Feasible Gain Sets

To find feasible βiωi and βpωi that correspond to communication delays, we need to

investigate root locus of the characteristic equation of the time-delay small-signal

model of the microgrid with respect to βiωi and βpωi.


Here, we define

Kiωi = βiωiKiωi (3.1)

and

Kpωi = βpωiKpωi (3.2)

where, Kiωi and Kpωi are the original load frequency control gains located in the

MGCC. The changeable parts are βiωi or βpωi located in each local DG controller.

The reason behind this is that the round-trip communication delays can be known

only in local DG controllers which receive load frequency controller outputs, while it

cannot been known in the MGCC which only generate the load frequency controller

outputs. Kiωi and Kpωi represent the equivalent gains of the secondary frequency

controller after gain schedulers are equipped in each LC. We consider gain schedulers

change only one gain variable, either βiωi or βpωi.

Then, the equalized load frequency controller has the following form:


s)(ω0 − ωi) (3.3)

or


s)(ω0 − ωi) (3.4)

The time-delay small signal model of the microgrid is built as follows.

EΔx = AΔx + AτΔx(t − τ) + Fr0 (3.5)

where E and A are system matrices, Aτ is a parameter matrix related to the delayed

state x(t − τ), and F is a parameter matrix. While most elements of the matrices are

still the same as the ones in Chapter 2, the original Kiωi is replaced by Kiωi or Kpωi

by Kpωi. The gain scheduler vector βiω and βpω denote four integral gain schedulers


and proportional gain schedulers, respectively.

Since the characteristic equation det(E − A − Aτ e−λτ ) = 0 is transcendent, it has

infinitely many roots. Therefore, we can only approximate its solution by computing

a reduced set of its roots. One effective technique to approximate its solution is to

approximate the roots of det(E − A − Aτ e−λτ ) = 0 by a finite element method [17].

In this method, a matrix M which has the following form is defined.

M =

⎡⎢⎢⎢⎢⎣

C ⊗ In

Aτ 0 A

⎤⎥⎥⎥⎥⎦ (3.6)

where ⊗ indicates Kronecker’s product, In is the identity matrix of order n, and C

is a matrix composed of the first N − 1 rows of C defined as follows:

C = −2DN/τ (3.7)

where DN is a Chebyshev’s differentiation matrix of dimension N + 1 × N + 1 (See

Appendix B for details). Then, the eigenvalues of M are an approximated spectrum

of the characteristic equation of (3.5).

For the simplicity of calculations, N is chosen as 2 for DN. Then, root locus of the

critical eigen pair are investigated for both βiωi and βpωi under different time delays

τ = {0.1, 0.2, 0.3}, shown in Fig 3.2 and Fig 3.3. From these two figures, for both βiωi

and βpωi, it can be seen that feasible gain sets that guarantee the microgrid system

stable become smaller as time delays inscrease in communication links.


−7 −6 −5 −4 −3 −2 −1 0 1 2−15

−10

−5

0

5

10

15

Real(1/s)

Imag

(rad

/s)

τ=0.1τ=0.2

τ=0.3

βiw

βiw

Figure 3.2: Root locus for βiω under different time delays (Arrows direct the increasinggains)

−5 −4 −3 −2 −1 0 1 2

−10

−5

0

5

10

Real(1/s)

Imag

(rad

/s)

τ=0.3

τ=0.1τ=0.2β

pω

βpω

βpω

=1.25

βpω

=1.05

βpω

=0.9

Figure 3.3: Root locus for βpω under different time delays (Arrows direct the increas-ing gains)


3.3.2 Feasible Gains with Respect to The Microgrid Perfor-

mance

To find relationships between the gain scheduler variables (βiωi and βpωi) and the

system performance of the microgrid, the following average squared error is defined

the cost function.

J = 1T

∫ T

t=0e2

i (t) (3.8)

where T is the simulation horizon, ei(t) = ωdi (t) − ω0

i (t) is the frequency error with

respect to the nominal frequency reference ω0i (t) for DG i. The nominal reference

ω0i (t) is the frequency of the original DG i when there are no communication delays

with initial gains Kiω = 60 and Kpω = 2, while ωdi (t) is the frequency of the DG i

when there are communication delays in the microgrid.

Time domain simulation trials of the microgrid in Matlab/SimPower R2007b are

carried out. The nominal references are the dynamics of the original microgrid without

communication delays with initial gains Kiω = 60 and Kpω = 2. The βiωi and βpωi

are investigated respectively. The performance cost curves with respect to βiω1 and

βpω1 for DG 1 are shown in Fig 3.4 and Fig 3.5, respectively. From Fig 3.4, it can

be seen that cost when τ = 0.1s and τ = 0.2s increase slowly, while the cost when

τ = 0.3s changes sharply. That means the microgrid is not stable any more when

τ = 0.3s. Also, in Fig 3.5, for βpω1, among all the costs for three different time delays,

cost when τ = 0.3s is still the largest.

The procedure of the gain scheduling approach includes the following steps:

• Step 1: Calculate the communication delay by comparing the time stamps of

two signals P SF ′refiT 2 and ωiT 1. The total time delay τi = T2 − T1, omitting the

time spent in calculating control outputs by the MGCC.

• Step 2: For a given cost index, by looking up in cost curves such as Fig 3.4 and


0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

βiω1

J

τ=0.1sτ=0.2sτ=0.3s

Figure 3.4: Cost curve with respect to βiω1 when Kpω = 2

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

βpω1

J

τ=0.1sτ=0.2sτ=0.3s

Figure 3.5: Cost curve with respect to βpω1 when Kiω = 60


Fig 3.5 built by the offline analysis, the local gain scheduler adjusts the value

of its gain scheduler variable according to the measured communication delay.

3.4 Simulations

The microgrid shown in Fig 2.1 is used to study the effect of different kinds of time-

varying delays on the performance of the microgrid with a load frequency controller.

Then, the effectiveness of the proposed gain scheduling approach is evaluated, with

comparison to fixed gain controllers.

The simulation platform for this microgrid is developed in Matlab/SimPower

R2007b environment. The load frequency controller gains are initially set to Kpω = 2

and Kiω = 60 based on the results of the small-signal analysis of the microgrid in

Chapter 2. The nominal frequency is chosen as ω0 = 377rad/s for the load frequency

control which is identical to the current Northern American standard frequency and

guarantees the easy reconnection of the microgrid to the utility main grid. The circuit

breaker at 120 kV and the 12.5 kV substation is considered to be initially open that

means the microgrid is islanded at t = 0s.

To investigate the impact of communication delays, the following scenarios are

considered.

• Case 1: there is a constant total communication delay τ = 0.1s in the microgrid;

• Case 2: there is a constant total communication delay τ = 0.2s in the microgrid.

The results of these two cases are shown in Fig 3.6 and Fig 3.7. Since four DGs in

the microgrid are identical, only the dynamics of DG1 are presented. From these

results, it can be noted that the frequency and real power of DG1 oscillate a lot in

the presence of time delays.


0 1 2 3 4 5375

376

377

378

379

ω (r

ad/s

)

(a) Frequency of DG1

0 1 2 3 4 50.2

0.3

0.4

0.5

Time (seconds)

P (p

u)

(b) Real power of DG1

Figure 3.6: Dynamic performance of DG1 with τ = 0.1s

0 1 2 3 4 5375

376

377

378

379

ω (r

ad/s

)

(a) Frequency of DG1

0 1 2 3 4 50.2

0.3

0.4

0.5

Time (seconds)

P (p

u)

(b) Real power of DG1

Figure 3.7: Dynamic performance of DG1 with τ = 0.2s


To evaluate the effectiveness of the proposed gain scheduling approach, the fol-

lowing cases are considered.

• Case 3: there is a time-varying delay following a uniform distribution with

τ ∈ {0, 0.1, 0.2} in the microgrid with βiωi gain-schedulers are installed in local

controllers of DGs.

• Case 4: there is a time-varying delay following a uniform distribution with

τ ∈ {0, 0.1, 0.2} in the microgrid with βpωi gain-schedulers are installed in local

controllers of DGs.

Simulation results of these two cases are presented from Fig 3.8 to Fig 3.12. A part

of the time-varying delay trajectory is shown in Fig 3.8. The uniform distribution

was used for estimating time delays in [47]. With comparison to other distributions,

the uniform distribution illustrates a worse communication delay process which may

happen in practice. We use it in this study to show the performance of the microgrid.

With this time-varying delay process, dynamic performances of DG1 are shown in

Fig 3.9. Apparently, with the βiω1 gain scheduler, the frequency dynamic of DG 1

behaves better than that without the gain scheduler. A part of the gain scheduling

process is shown in Fig 3.10. As shown in Fig 3.11 and Fig 3.12, similar conclusions

also are made for the DG 1 with the βpω1 gain scheduler.

3.5 Summary

A gain scheduling approach has been presented for compensating the impact of com-

munication delays on the load frequency control performance of an islanded microgrid

in this chapter. This approach consists of two steps. For the first step, feasible gain

sets with respect to a given cost function are found by offline small-signal analysis.

For the second step, via GPS embedded in PMUs and gain schedulers, each local


0 20 40 60 80 100−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Time (× 1e−5 seconds)

τ (s

econ

ds)

Figure 3.8: The dynamic of the time-varying delay (only showing first 100 samples)

0 1 2 3 4 5375

375.5

376

376.5

377

377.5

378

378.5

379

Time (seconds)

ω (r

ad/s

)

Frequencies of DG1

without a gain schedulerwith a gain scheduler

Figure 3.9: Dynamic performances of DG1 with a βiω1 gain-scheduler


0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1


β iw1

Figure 3.10: The dynamic of the βiω1 gain-scheduler (only showing first 100 samples)

0 1 2 3 4 5375

375.5

376

376.5

377

377.5

378

378.5

379

Time (seconds)

ω (r

ad/s

)

Frequencies of DG1

without a gain schedulerwith a gain scheduler

Figure 3.11: Dynamic performances of DG1 with a βpω1 gain-scheduler


0 20 40 60 80 1000.4

0.5

0.6

0.7

0.8

0.9

1

1.1


β pw1

Figure 3.12: The dynamic of the βpω1 gain-scheduler (only showing first 100 samples)

controller can calculate current communication delays and then schedule the corre-

sponding load frequency controller gains. Simulations of an islanded microgrid with

4 inverter-based DGs under different time-delay cases have verified the effectiveness

of the proposed gain scheduling method.

Chapter 4

Stability Analysis of Load Frequency

Control over Cognitive Radio Networks in

Largely Interconnected Power Grids

4.1 Introduction

In the previous two chapters, we investigate the analysis and control the effects of

communication delays on the miocrogrid load frequency control (LFC). In the fol-

lowing two chapters, the impacts of communication failures on largely interconnected

power grids are analyzed. Two specific situations that can result in communication

failures in smart grids are considered. One of them is the utilization of cognitive

radio (CR) networks to support smart grid communication which is presented in this

chapter. The other is the denial of service (DoS) attack to smart grid communication

links which will be presented in Chapter 5.

In this chapter, the stability of the LFC of a largely interconnected power system

for which CR networks are used as the communication infrastructure is analyzed.

For this purpose, a new switched power system model is proposed for the LFC of the

power system by modeling the CR network as an On-Off switch with sojourn times.

51

CHAPTER 4. STABILITY ANALYSIS OF LFC OVER CR NETWORKS 52

Sufficient conditions are obtained for the stability of the LFC of the power system

with two different kinds of CR networks. Simulation results show the effect of CR

networks on the dynamic performance of the LFC of the power system and illustrate

the usefulness of the developed sufficient conditions in the design of CR networks in

smart grids.

The rest of this chapter is organized as follows. In Section 4.2, the CR network

issues in smart grid applications are introduced. In Section 4.3, a new switched

system model is proposed to investigate the effect of CR networks on the stability

of the LFC of a largely interconnected power grid. In Section 4.4, the stability of

the LFC of the power grid over CR networks is studied for both deterministic and

stochastic sojourn time situations. In Section 4.5, a two-area power system is used

for simulation studies. Finally, Section 4.6 concludes the chapter.

4.2 Cognitive Radio Networks in Smart Grids

A smart grid integrates advanced two-way communication networks and advanced

intelligent computing technologies into current power systems, from large-scale gen-

eration through delivery units to electricity consumers [2, 48, 49]. The applications

of a smart grid include wide-area monitoring, control and protection (WAMCP), dis-

tributed generation management, advanced metering infrastructure (AMI), real-time

pricing, etc. [1, 50–52]. To support these applications, there are several unique chal-

lenges to be addressed for smart grid communications [53–56].

Limited bandwidth: Since WAMCP and AMI involve tremendous amounts of in-

formation exchange over wide geographical areas, it needs large bandwidths for both

data transmission and collection.

Interference: As most of AMI communication architectures are normally formed a-

mong smart meters for data routing in the 2.4 GHz industrial, scientific, and medical


(ISM) band, signal interferences will be severe among these types of radio systems.

Inter-operability: Last but not least, because a smart grid communication archi-

tecture consists of wide area network (WAN), neighborhood area network (NAN)

and home area network (HAN), this heterogeneous network architecture requires the

capability to coordinate communications within each subarea and between different

areas. However, most of the traditional communication technologies are infeasible to

meet all these requirements.

Due to its great potentials to enhance the overall performance of data commu-

nications with its dynamic and adaptive spectrum management capabilities, the CR

networking technology has been increasingly considered as the networking and com-

munication infrastructure for smart grids [55–57]. In view of the fact that a large por-

tion of the licensed radio spectrum remains severely under-utilized, the CR technology

is proposed to achieve the efficient usage of the assigned radio spectrum [47, 58–60].

In a CR network, there are two kinds of users. Primary users (PUs) are the users

who are licensed with certain bands of the current spectrum, while secondary users

(SUs) do not have the licenses for the utilization of those spectrum bands. Howev-

er, SUs can opportunistically sense and identify the unused channels in the licensed

spectrum. Based on the sensed results, SUs are able to use the available channels,

coordinate the spectrum access with other users, and return the channel back to PUs

when PUs reclaim the spectrum usage right. With this capability of the dynamic

and opportunistic spectrum allocation, CR networks can increase spectrum efficien-

cy, enable large-scale different spectrum regulations, and coordinate radio spectrum

sharing among different area networks in smart grids.

Although the CR networking technology has great potentials to address the unique

challenges for smart grid communications in comparison with many other networking

technologies, it brings in a new problem. Specifically, a SU of a CR network has to

be squeezed out from the channel that it is using when a PU reclaims to use the


spectrum and this may occur in a randomized fashion. The random interruption of

SU traffics will unavoidably cause data packet losses and delays for SU data. Besides

the traffic interruption resulted from PUs, data packets could also be lost due to

traffic congestions resulted from other SUs who also want to send data at the same

time. These lost data packets can be of any natures: control commands from control

centers to substations, sensed data from remote terminal units (RTUs), real-time

pricing information between utilities and customers, etc.. The loss of these data may

lead to very severe and adverse effects on the monitoring and control processes of a

smart grid. In [28], it is emphasized that communication failures of a power grid may

cause very serious problems for both system operation and control.

Therefore, it is very important to deal with the above problem and to understand

the effects of the random interruptions of SU traffics in CR networks on the stability

and performance of the smart grid monitoring and control processes. To our best

knowledge, there is no existing work to address this issue. In this chapter, we study

this problem and investigate stabilities of the LFC of a largely interconnected power

grid for which CR networks are used as the infrastructure for the aggregation and

communication of both system-wide information and local measurement data.

4.3 Modeling of The LFC over A Cognitive Radio

Network in A Largely Interconnected Power

System

The LFC over a CR network in a two-area power system is shown in Fig 4.1. In this

LFC, there are two data exchange loops. One of them is the feed-forward loop in

which control centers send control signals to RTUs. The other is the feedback loop

where measurement signals are transmitted from RTUs to the control centers over the


Figure 4.1: Two-area power system over cognitive radio (CR) networks

CR wireless network. It can been noticed that the LFC is a typical networked control

system (NCS), of which the control loop is closed via communication links [61–64].

In this section, a new On-Off switch model of the CR network for smart grid

communications is proposed. By using this On-Off switch model, a linear switched

system is proposed to integrate the dynamic of the CR network in the cyber layer

into the physical power system in the smart grid.

4.3.1 The Model of Cognitive Radio Networks

As shown in Fig 4.2, we consider a licensed spectrum band consisting of N non-

overlapping channels in the CR network used by the smart grid. Both PUs and SUs in

this CR network are operated synchronously in a time-slotted fashion. In this chapter,

we consider the situation in which each cognitive user can sense only one channel at

each time slot. The availability of each channel is modeled as a 2-state Markov chain

in the literature [47, 59, 60]. However, in terms of the system performance of the


Figure 4.2: The cognitive radio channel illustration

Figure 4.3: The proposed On-Off cognitive channel model

smart grid, we should take account of not only the state transitions, but also the

time staying in each state. From each SU point of view, the sensed channel state is

either free or occupied by other users. This means the communication channel for

data exchange is either ON or OFF. Thus, we model the communication channel as

an ON-OFF switch with sojourn times, shown in Fig 4.3. A sojourn time τi is a

time interval the communication channel continuously stays in a state, either ON or

OFF [65]. The channel state at kth time instant is denoted by θk ∈ Θ = {0, 1}, where

Θ = {0, 1} is the state space of θk, 0 denoting OFF state of the channel, 1 denoting

ON state.


4.3.2 The Switched System Model for The LFC over A Cog-

nitive Radio Network

The operation of the frequency control in power systems is fundamental in determin-

ing the way in which the frequency will change when load changes happen [8,32,66].

LFC mainly keeps the frequency of the power system at a nominal value (i.e 60Hz)

by adjusting power generation set points. For the LFC of a largely interconnected

power system, the power system is decomposed into several control areas which are

interconnected by high voltage tie-lines. In each control area, it comprises a group of

generators and a number of loads. Commonly, the generators are represented equiva-

lently by one single machine and the loads by one single load. Furthermore, although

power systems are usually non-linear, linearized models are used because the LFC

operation only involves relatively small disturbances.

The control structure of an equivalent linearized model in the ith control area

is shown in Fig 4.4. A non-reheat steam turbine is considered. When the governor

senses the frequency changes Δfi, it adjusts the valve position ΔPvi. By doing this,

the input of steam flowing into the turbine is regulated and thus the mechanical power

ΔPmiis controlled. As a result, the frequency is kept constant. The governor and

turbine compose the primary frequency control of this generating unit. However, the

system frequency is usually not able to be restored to the nominal value (i.e. 60Hz) by

only using the primary control. A secondary frequency controller K(s) is necessary to

adjust the load reference set point ΔPcifor the turbine to make the power generation

ΔPmitrack the load changes ΔPLi

and restore the system frequency.

The turbine dynamic is described by

ΔPmi= − 1

Tchi

ΔPmi+ 1

Tchi

ΔPvi(4.1)


where, ΔPmiis the generator mechanical power deviation, ΔPvi

is turbine valve po-

sition deviation, and Tchiis the time constant of turbine i.

The governor dynamic is described by

ΔPvi= − 1

RiTgi

Δfi − 1Tgi

ΔPvi+

1Tgi

ΔPci(4.2)

where, Δfi is the frequency deviation of area i, ΔPciis the load reference set-point,

Tgiis the time constant of governor i, and Ri is the speed droop coefficient.

The overall load-generation dynamic is described by

Δfi = − Di

2HiΔfi + 1

2HiΔPmi

− 12Hi

ΔP itie − 1

2HiΔPLi

(4.3)

where, ΔP itie is the net tie-line power flow in area i, ΔPLi

is the load deviation, Hi is

the equivalent inertia constant of area i, and Di is the equivalent damping coefficient

of area i.

The dynamic of the net tie-line power flow dynamic is

ΔP itie =

N∑j=1,j �=i

2πTij(Δfi − Δfj) (4.4)

where Tij synchronizing power coefficient, and Δfj is the of area j.

Furthermore, we assume there are N interconnected areas in the power system.

We can write the state space model of the above dynamics for LFC in area i as follows:

xi = Aiixi + Biui +N∑

j=1,j �=i

Aijxj + FiΔPLi, xi(0) = x0 (4.5)

where

xi =[

Δfi ΔPmiΔPvi

ΔP ijtie

]T

; ui = ΔPci;


Figure 4.4: The block diagram of the control area i

Aii =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

− Di

2Hi

12Hi

0 − 12Hi

0 − 1Tchi

1Tchi

0

− 1RiTgi

0 − 1Tgi

0

N∑j=1,j �=i

2πTij 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

; Aij =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0

0 0 0 0

0 0 0 0

−2πTij 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

;

Bi =[

0 0 1Tgi

0

]T

; Fi =[

− 1Mi

0 0 0

]T

.

For the whole multi-area power system, a centralized linear time-invariant (LTI)

interconnected model is given by:

x = Acx + Bcu + FΔPL, x(0) = x0 (4.6)

where

x =[

x1 x2 · · · xN

]T

; u =[

u1 u2 · · · uN

]T

;


ΔPL =[

ΔPL1 ΔPL2 · · · ΔPLN

]T

;

Ac =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

A11 A12 · · · A1N

A21 A22 · · · A2N

... ... . . . ...

AN1 AN2 · · · ANN

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

; Bc = diag

{B1 B2 · · · BN

}T

;

The above state-space model can be written in a more convenient form as the

steady state is denoted by xss [67]:

x = Acx + Bcu, x(0) = −xss

Due to the CR network has been modeled as an ON-OFF switch in the slotted

time fashion (inherently a discrete-time model), the following discrete-time model of

the power system is considered:

x(k + 1) = Ax(k) + Bu(k) (4.7)

where, A = eAch, B =∫ h

0 eAcτ Bcdτ , h is the sampling period.

The above power model is under the assumption that the communication channel

between the control center and RTUs is perfect. However, as we discussed in the

previous section, the fact that PUs reclaim the channel usages will cause packet

losses when data are transferring within SUs. The channel state at the kth time slot

is denoted by θk ∈ Θ = {0, 1}, where 0 denoting OFF state of the channel (packets

are lost), 1 denoting ON state (packets are received successfully). Therefore, a new

switched system model is proposed for the LFC over CR networks in smart grids.

Zero-order-holds (ZOHs) are used in the control center [68]. Considering the time


interval l ∈ [tk, tk+1), where tk, tk+1 are two consecutive state jump instants, the state

feedback controller for the LFC becomes the following two modes:

u(l) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

Kx(l), θtk= 1

Kx(tk), θtk= 0

(4.8)

Plugging the above state feedback controller into the power system (4.7), the dynamics

of the closed-loop power system have the following linear switched system form during

the time interval l ∈ [tk, tk+1):

x(l + 1) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

(A + BK)x(l), θtk= 1

Ax(l) + BKx(tk), θtk= 0

(4.9)

From the iterative deduction of the power system (4.7), we can get the following

equations of x(l) for l ∈ [tk, tk+1), based on x(tk).

x(1) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

(A + BK)l−tkx(tk), θtk= 1

(Al−tk +l−tk−1∑

r=0ArBK)x(tk), θtk

= 0(4.10)

The initial instant is denoted by t0 and the initial state x(t0) and θt0 are the initial

conditions. Until tk+1, the sojourn time sequence is {τ1, τ2, · · · , τi, · · · , τk+1}. In view

of the fact that PUs opportunistically reclaim the communication channel back from

SUs in the CR network, the sojourn time τi = ti+1 − ti is a time-varying variable and

is independent of other sojourn times within the sequence {τ1, τ2, · · · , τi, · · · , τk+1}.

Correspondingly, the closed-loop power system is a time-varying linear switched sys-

tem.


4.4 Stabilities of LFC over Cognitive Radio Net-

works

In the previous section, we propose a time-varying linear switched system model for

the closed-loop power system over a CR network. In this section, the stabilities of

the LFC over CR networks are going to be studied and sufficient conditions will be

given for both the asymptotical stability and the mean-square stability of the power

system under two different kinds of CR networks. Here, we consider the cases that

sojourn time variables {τ1, τ2, · · · , τi, · · · , τk+1} are independent to each other.

4.4.1 Asymptotical Stability for Arbitrary but Bounded So-

journ Times

Definition. 4.4.1 For arbitrary sojourn times τi ∈ [τmin, τmax], i ∈ {0, 1, · · · , k + 1},

the state trajectory x(l) in equation (4.10) with initial conditions x(t0) = x0 and

θt0 = θ0 ∈ Θ = {0, 1} is globally asymptotically stable if for any ε > 0, there exists

a β > 0, whenever ||x0|| < β, xl satisfies ||x(l, t0, x0)|| < ε for any l > t0, and

liml→∞ ||x(l, t0, x0)|| = 0.

Theorem 4.4.1. The power system (4.10) is globally asymptotically stable for arbi-

trary sojourn time τi ∈ [τmin, τmax], i ∈ {0, 1, · · · , k + 1} if all the eigenvalues of the

matrices (Aτi +τi−1∑r=0

ArBK) are inside the unity circle

Proof. When a sojourn time variable follows an arbitrary but bounded distribution,

it changes randomly in a give range such as τi ∈ [τmin, τmax], i ∈ {0, 1, · · · , k +

1}. Without loss of generality, we consider the initial conditions x(t0) = x0

and θt0 = 1 which means the communication channel is ON and packets are

successfully transmitted. On the time interval l ∈ [tk, tk+1), the sojourn time

sequence is {τ1, τ2, · · · , τi, · · · , τk, τk+1}, and the corresponding state sequence is


{1, 0, 1, 0, · · · , 1, 0}. On the time interval l ∈ [tk, tk+1), θtk= 0, the system (4.10)

has the following state response:

x(l) = (Al−tk +l−tk−1∑

r=0ArBK)x(tk)

Using the system equation (4.10), x(tk) has the following response:

x(tk) = (A + BK)τkx(tk−1)

And,

x(tk−1) = (Aτk−1 +τk−1∑r=0

ArBK)x(tk−2)

By this deduction, the state x(l) has finally the following form:


r=0ArBK)(A + BK)τk(Aτk−1 +

τk−1∑r=0

ArBK) · · · (A + BK)τ1x0

(4.11)

The state response norm is given as follows,

||(Al−tk +l−tk−1∑

r=0ArBK)(A + BK)τk(Aτk−1 +

τk−1−1∑r=0

ArBK) · · · (A + BK)τ1x0||

≤ λmax{Aτmax +τmax−1∑

r=0ArBK}||(A + BK)τk · · · (A + BKτ1)||||x0||.

(4.12)

If all the eigenvalues of both (A+BK)τi and (Aτi +τi−1∑r=0

ArBK) are inside the unity

circle for τi ∈ [τmin, τmax], i ∈ {0, 1, · · · , k+1}, the convergence of the matrix products

is guaranteed. In view of the fact that the original closed-loop system A + BK

is stable, all the eigenvalues of the matrix (A + BK)τi are inside the unity circle.

Therefore, the system (4.10) is globally asymptotically stable if all the eigenvalues of

the matrices (Aτi +τi−1∑r=0

ArBK) are inside the unity circle for τi ∈ [τmin, τmax], i ∈{0, 1, · · · , k + 1}.


Thus, sufficient conditions for the globally asymptotical stability of the system

(4.10) are obtained.

With this theorem, we are able to find the largest interval [τmin, τmax] for τi, below

which the globally asymptotical stability of the system (4.10) can be guaranteed. By

calculating the maximum eigenvalues for (Aτi +τi−1∑r=0

ArBK) with τi, the τmax will

be the values of τi that (Aτi +τi−1∑r=0

ArBK) have the maximum eigenvalues outside of

unity circle for the first time.

4.4.2 Mean-square Stability for Random Sojourn Times with

Independent Identical Distribution

In this section, we study the stochastic stability (mean square stability) of the power

system (4.10) with random sojourn times {τ1, τ2, · · · , τi, · · · , τk, τk+1} which follow

the identical probability density function (p.d.f) p(τi) for all i ∈ {1, 2, · · · , k + 1}.

Definition. 4.4.2 For i.i.d sojourn times which follow the probability density

function (p.d.f) p(τi), for all i ∈ {1, 2, · · · , k + 1}, the state trajectory x(l) in system

(4.10) with initial conditions x(t0) = x0 and θt0 = θ0 ∈ Θ = {0, 1} is mean square

stable if xl satisfies liml→∞ E{||x(l, t0, x0||2)} = 0.

We give sufficient conditions under which the system (4.10) is mean square stable

with random sojourn times.

Theorem 4.4.2. The system (4.10), with sojourn times τi, i ∈ {1, 2, · · · , k + 1},

which follow the identical probability density function (p.d.f) p(τi), is mean square

stable, if the following inequalities hold for all i ∈ {1, 2, · · · , k + 1}:

• (a) The expected maximum singular value of the matrices Ψ(τi) is convergent,

that is∞∑

τi=1p(τi)σmax(Ψ(τi)) < ∞, where Ψ(τi) = (Aτi +

τi−1∑r=0

ArBK);


• (b) The expected maximum singular value of the matrices Υ(τi) is convergent,

that is∞∑

τi=1p(τi)σmax(Υ(τi)) < ∞, where Υ(τi) = (A + BK)τi;

• (c)∞∑

τi=1p(τi)σmax(Υ(τi))2 < ∞;

• (d)∞∑

τi=1p(τi)σmax(Ψ(τi))2 < ∞, where, σmax(∗) denotes the maximum singular

value of the matrix ∗.

Proof. We consider the initial conditions x(t0) = x0 and θt0 ∈ Θ = {0, 1}. For each

initial state case, the final state can be either θtk= 0 or θtk

= 1. The sojourn time

sequence during the time interval [t0, tk+1) is denoted by {τ1, τ2, · · · , τi, · · · , τk, τk+1}.

On the last state interval [tk, tk+1], for any l ∈ [tk, tk+1), the state x(l) of the system

(4.10) has the following possible forms.

(i) When the initial state of communication channel is OFF and the finial state is

OFF, the x(l) has the following form.


r=0ArBK)(A+BK)τk · · · (A+BK)τ2(Aτ1 +

τ1−1∑r=0

ArBK)x0 (4.13)

Let Φ(τk) = (A + BK)τk · · · (A + BK)τ2(Aτ1 +τ1−1∑r=0

ArBK). The expectation of the

square norm of x(l) is

E {||x(l)||2} = E

{xT

0 [Φ(τk)T (Al−tk +l−tk−1∑

r=0ArBK)T (Al−tk +

l−tk−1∑r=0

ArBK)Φ(τk)]x0

}

≤ E{σmax(Aτk+1 +τk+1−1∑

r=0ArBK)}E

{Φ(τk)T Φ(τk)

}||x0||2

(4.14)

where, σmax(∗) denotes the maximum singular value of the matrix ∗.


Due to

E{Φ(τk)T Φ(τk)

}≤ E {||(A + BK)τk ||2} · · · E {||(A + BK)τ2||2}

E

{||Aτ1 +

τ1−1∑r=0

ArBK||2}

, we have the following inequality.

E {||x(l)||2} ≤ E

{σmax(Aτk+1 +

τk+1−1∑r=0

ArBK)}

E {||(A + BK)τk ||2}

· · · E {||(A + BK)τ2||2} E

{||Aτ1 +

τ1−1∑r=0

ArBK||2}

||x0||2

Since all the sojourn times τi, i ∈ {0, 1, · · · , k + 1} follow the identical p.d.f p(τi), if

the following inequalities hold true:

E {σmax(Ψ(τi))} =∞∑

τi=1p(τi)σmax(Ψ(τi)) < ∞ (4.15)

where, Ψ(τi) = (Aτi +τi−1∑r=0

BK), p(τi) is the probability density function of τi,

and,

E{||(A + BK)τi||2

}=

∞∑τi=1

p(τi)||Υ(τi)||2 < ∞ (4.16)

where, Υ(τi) = (A + BK)τi,

and,

E

{||Aτi +

τi−1∑r=0

BK||2}

=∞∑

τi=1p(τi)||Ψ(τi)||2 < ∞ (4.17)

then, E {||x(l)||2} < ∞ and∞∑

l=0E {||x(l)||2} < ∞. Therefore, system (4.10) is mean

square stable.

(ii) If the initial state of communication channel is OFF and the finial state is


ON, then

x(l) = (A + BK)1−tk(Aτk +τk−1∑r=0

BK) · · · (A + BK)τ2(Aτ1 +τ1−1∑r=0

ArBK)x0 (4.18)

Following the same procedure as (i), we get the same results for (ii), except the

following fact

E {σmax(Υ(τi))} =∞∑

τi=1p(τi)σmax(Υ(τi)) < ∞ (4.19)

p(τi) is the probability density function of τi.

(iii) If the initial state of the communication channel is ON and the finial state is

OFF, then


r=0ArBK)(A+BK)τk · · · (Aτ2 +

τ2−1∑r=0

ArBK)(A+BK)τ1x0 (4.20)

Following the same procedure as (i), we get the same results as (i).

(iv) If the initial state of the communication channel is ON and the finial state is

ON, then

x(l) = (A+BK)1−tk(Aτk−1 +τk−1∑r=0

ArBK) · · · (Aτ2 +τ2−1∑r=0

ArBK)(A+BK)τ1x0 (4.21)

Following the same procedure as (i), we get the same results as (ii).

If Ψ(τi) and Υ(τi) are matrices with the ranks are m and n respectively, their

singular values are denoted by σj(Ψ(τi)) and σj(Υ(τi)). According to the properties

of the singular values [69], ||Ψ(τi)||2 =m∑

j=1σ2

j (Ψ(τi)),and ||Υ(τi)||2 =n∑

j=1σ2

j (Υ(τi)).

Thus, the condition (4.16) is equivalent to,

E {||(A + BK)τi||2} =∞∑

τi=1p(taui)||Υ(τi)||2 =

∞∑τi=1

p(τi)n∑

j=1σ2

j (Υ(τi))

≤ n∞∑

τi=1p(τi)σmax(Υ(τi))2 < ∞

(4.22)


and the condition (4.17) is equivalent to

E

{||Aτi +

τi−1∑r=0

BK||2}

|| =∞∑

τi=1p(τi)||Ψ(τi)||2 =

∞∑τi=1

p(τi)m∑

j=1σ2

i (Ψ(τi))

≤ m∞∑

τi=1p(τi)σmax(Ψ(τi))2 < ∞

(4.23)

In summary, if the inequalities (4.16), (4.19), (4.22), (4.23) hold, the system (4.7)

is mean square stable. This completes the proof.

A maximum singular value is usually used as an important indicator to the ro-

bust stability of linear systems with uncertainties [70, 71]. Therefore, the expected

maximum singular values of the matrices can be used as the upper bounds for the

stochastic stability of the system (4.10).

4.5 Simulations

In this section, a two-area power system model is used to evaluate the effects of CR

networks on the system performance. For stochastic sojourn time cases, both uniform

and geometric p.d.f sojourn time processes are simulated to evaluate their influences

to the dynamic performance of the power system. The two processes are firstly evalu-

ated under two different initial frequency changes resulted from load changes. Then,

they are investigated under two different tie-line power situations. For the arbitrary

sojourn time case, the largest sojourn time interval below which the asymptotical

stability can be guaranteed is calculated. For the stochastic sojourn time case, the

expected maximum singular values are calculated for the sojourn times of CR net-

works follow a uniform probability distribution. All the numerical simulations are

run in Matlab R2012a in a computer with 2.66GHz CPU and 8.00GB RAM. In this

study,100MVA is chosen as the base unit for per unit (pu) calculations.


Figure 4.5: Two-area power system

The two-area power system is shown in Fig 4.5 and its parameters are shown in

Appendix C. Its state space model is given as follows.

x = Acx + Bcu (4.24)

where

where

x =[

ΔPtie Δf1 ΔPm1 ΔPv1 Δf2 ΔPm2 ΔPv2

]T

; u =[

ΔPc1 ΔPc2

]T

;


Ac =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0.545 0 0 −0.545 0 0

−5 −0.07 5 0 0 0 0

0 0 −2.5 2.5 0 0 0

0 −5.21 0 −12.5 0 0 0

6 0 0 0 −0.05 6 0

0 0 0 0 0 −2.78 2.78

0 0 0 0 −6.94 0 −16.67

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

;

Bc =

⎡⎢⎢⎢⎢⎣

0 0 0 12.5 0 0 0

0 0 0 0 0 0 16.67

⎤⎥⎥⎥⎥⎦

T

.

A Linear Quadratic Regulator (LQR) is designed for this power system u = Kx,

the control gains are

K =

⎡⎢⎢⎢⎢⎣

−0.1915 0.4640 1.1086 0.4379 −0.1445 −0.3362 −0.0417

−3.4763 −0.6687 −0.5064 −0.0590 0.9898 1.6769 0.4275

⎤⎥⎥⎥⎥⎦

(4.25)

Firstly, the effects of a CR network with uniform p.d.f. sojourn times on the power

system performance are evaluated. The following mean square error (MSE) of the

state of the power system is used as the metric of its dynamic performance:

MSE(k) = 1N

N∑i=1

(x(k) − x0(k))T (x(k) − x0(k)) (4.26)

where, x0(k) is the original power system state at kth time instant without any

disturbances, and x(k) is the system state when a CR network is used, and N denotes


how many times the simulations run. Here, the simulations run 500 times and the

time horizon of each simulation is 5 seconds. In order to evaluate different load

changes and tie-line power flows in the power system, four different initial conditions

are considered:

• Case 1: Δf1(0) = 1.5Hz in Area 1 and Δf2(0) = 0.5Hz in Area 2;

• Case 2: Δf1(0) = 2Hz in Area 1 and Δf2(0) = 1Hz in Area 2;

• Case 3: ΔP 12tie(0) = 0.5(pu);

• Case 4: ΔP 12tie(0) = 1(pu).

The MSEs of the states of the power system with random sojourn times which

follow uniform p.d.f for the four cases are shown from Fig 4.6 to Fig 4.9. By comparing

these Fig 4.6 and Fig 4.7, it can be seen that the system performance becomes much

worse when the load changes in two areas are bigger. Also, for both cases, the system

performance is the worst over the CR network with the biggest τmax. When tie line

power are considered, the comparison between the results of Fig 4.8 and Fig 4.9 shows

that the system performance is worse when the larger tie-line power are transferring

between area 1 and area 2. Moreover, for these two tie-line power cases, the system

performance is the worst over the CR network with the biggest τmax. According to

these results, with a given maximum mean square error that a power system can

endure, the maximum sojourn time that a CR network should not be exceeded can

be estimated when the CR network is designed for a smart grid.

Then, the effects of a CR network with geometric p.d.f. sojourn times on the power

system performance are evaluated. The MSEs of the states of the power system with

random sojourn times which follow geometric p.d.f for the four cases are shown from

Fig 4.10 to Fig 4.13, while the parameter p denotes the success probability of the

geometric p.d.f. When load changes are considered, the comparison between the


0 1 2 3 4 50

0.005

0.01

0.015

0.02

0.025

0.03

Time (s)

Mea

n sq

uare

err

ors

τmax

=0.05s

τmax

=0.1s

τmax

=0.15s

Figure 4.6: Mean square errors of the power system states with uniform p.d.f sojourntimes in Case 1

0 1 2 3 4 50

0.01

0.02

0.03

0.04

0.05

0.06

Time (s)

Mea

n sq

uare

err

ors

τmax

=0.05s

τmax

=0.1s

τmax

=0.15s



0 1 2 3 4 50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Time (s)

Mea

n sq

uare

err

ors

τmax

=0.05s

τmax

=0.1s

τmax

=0.15s


0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

Time (s)

Mea

n sq

uare

err

ors

τmax

=0.05s

τmax

=0.1s

τmax

=0.15s



0 1 2 3 4 50

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Time (s)

Mea

n sq

uare

err

ors

p=0.4p=0.6p=0.8

Figure 4.10: Mean square errors of the power system states with geometric p.d.fsojourn times in Case 1

results of Fig 4.10 and Fig 4.11 tells that the system performance becomes much

worse when the load changes in two areas of the power system are bigger. Also, it

can be seen from the two figures that the system performance is the worst over the CR

network with the smallest success probability of the geometric p.d.f. (p = 0.4). When

tie line power changes are taken into accounts, by comparing Fig 4.12 and Fig 4.13,

it is shown that the system performance is worse when the larger tie-line power are

transferring between area 1 and area 2. Furthermore, for these two tie-line power

cases, the system performance is the worst over the CR network with the smallest

success probability of the geometric p.d.f. (p = 0.4). According to these results, with

a given maximum mean square error that a power system can endure, the smallest

successful transmission rate that a CR network should be reached can be estimated

when the CR network is designed for a smart grid.


0 1 2 3 4 50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Time (s)

Mea

n sq

uare

err

ors

p=0.4p=0.6p=0.8


0 1 2 3 4 50

0.01

0.02

0.03

0.04

0.05

0.06

Time (s)

Mea

n sq

uare

err

ors

p=0.4p=0.6p=0.8



0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

Time (s)

Mea

n sq

uare

err

ors

p=0.4p=0.6p=0.8


Furthermore, the maximum sojourn time is calculated for the arbitrary but bound-

ed sojourn time case according to Theorem 4.4.1. Maximum eigenvalues with different

sampling periods are listed in Table 4.1. From this table, we can obtain the maxi-

mum sojourn times below which the asymptotical stability of the power system are

guaranteed with different sampling periods, by checking the first time when maxi-

mum eigenvalues are outside unity circle which are indicated as bolded and colored

numbers in the table.

Finally, we consider a stochastic sojourn time case. The stochastic sojourn times

are assumed to follow the uniform p.d.f within [τmin, τmax]. This kind of sojourn times

has been used to model the sojourn times in CR networks [60]. The relationships be-

tween the expected maximum singular values and the maximum sojourn time intervals

with different sampling periods have been shown in Table 4.2 and Table 4.3. These

tables are able to give suggestions on how to design the CR networks to guarantee


Table 4.1: The maximum eigenvalues of the matrices λ(Ψ(τ)) with different samplingperiods Ts

τ (time steps) λ(Ψ(τ)) (Ts = 0.05s) λ(Ψ(τ)) (Ts = 0.1s) λ(Ψ(τ)) (Ts = 0.5s) λ(Ψ(τ)) (Ts = 1s)

1 0.9092 + 0.1409i 0.8252 + 0.2636i -0.0860 + 0.7328i -0.5639 + 0.1830i

2 0.7998 + 0.2558i 0.5905 + 0.4278i -0.4829 + 0.1912i 0.2758 + 0.2706i

3 0.6754 + 0.2961i -0.6043 0.3125 + 0.3354i -0.0851 + 0.2006i

4 -1.3261 -0.9937 0.2788 + 0.3278i 0.0219 + 0.1312i

5 -2.0714 -1.8570 -0.2343 + 0.2897i 0.0334 + 0.0996i

6 -2.6570 -2.7057 0.3935 -0.0790

7 -3.6432 -3.3791 0.4433 0.0328 + 0.0325i

8 -4.7127 -3.9666 0.5127 -0.0488

9 -5.8693 -4.4563 0.5803 0.0364

10 -7.0743 -4.8474 0.6420 0.0783

11 -8.3116 -5.1475 0.7023 0.0256

12 -9.5671 -5.3701 0.7640 -0.1077

13 -10.8281 -5.5323 0.8264 0.0127

14 -12.0836 -5.6525 0.8887 -0.1372

15 -13.3242 -5.7484 0.9507 -0.1519

16 -14.5420 -5.8357 1.0127 -0.1666

the stochastic stability of the LFC of smart grids, according to the corresponding re-

lationships between the maximum sojourn time interval and the expected maximum

singular value.

4.6 Summary

In this chapter, we have studied the stability of the LFC of smart grids for which

cognitive radio (CR) networks are used as the communication and networking infras-

tructure. By modeling a CR network as an On-Off switch with sojourn times, a new

switched power system model has been proposed for the LFC of a smart grid. The

stability of the LFC of the smart grid has been studied for two main types of CR

networks: 1) the sojourn times are arbitrary but bounded, and 2) the sojourn times

follow an independent and identical distribution (i.i.d) process. For the first type of


Table 4.2: The maximum singular values σmax(Ψ(τ)) with the maximum sojourntimes τmax under sampling period Ts

τmax σmax(Ψ(τ)) σmax(Ψ(τ)) σmax(Ψ(τ)) σmax(Ψ(τ))time steps h = 0.05 h = 0.1 h = 0.5 h = 1

1 1.7391 2.1950 2.7718 1.2238

2 11.7573 9.6373 2.6534 1.3902

3 15.2699 12.1562 2.7059 1.2699

4 17.2216 13.5203 2.5066 1.2856

5 18.5933 14.4820 2.4528 1.2717

6 19.7057 15.2893 2.4636 1.2775

7 20.6924 16.0455 2.4849 1.2872

8 21.6170 16.7984 2.5049 1.2983

9 22.5131 17.5686 2.5339 1.3148

10 23.4002 18.3613 2.5719 1.3307

Table 4.3: The maximum singular values σmax(Υ(τ)) with the maximum sojourntimes τmax under sampling period Ts

τmax σmax(Υ(τ)) σmax(Υ(τ)) σmax(Υ(τ)) σmax(Υ(τ))time steps h = 0.05 h = 0.1 h = 0.5 h = 1

1 1.7391 2.1950 2.7718 1.2238

2 1.9304 2.3592 2.0105 0.9436

3 2.0618 2.4618 1.8759 0.7941

4 2.1604 2.5110 1.5547 0.6828

5 2.2368 2.5116 1.4447 0.5965

6 2.2953 2.4707 1.2699 0.5270

7 2.3380 2.3975 1.1759 0.4701

8 2.3663 2.3022 1.0582 0.4227

9 2.3814 2.1949 0.9824 0.3828

10 2.3844 2.0856 0.8987 0.3490


CR networks, sufficient conditions have been derived for ensuring the asymptotical

stability of the LFC of the smart grid. For the later one, sufficient conditions have

been found to guarantee the mean-square stability of the LFC of the smart grid.

Simulation results show how the conflict between PU and SU traffics affects the dy-

namic performance of the smart grid. They also show that the degraded dynamic

performance and these developed sufficient conditions can be used to estimate the

maximum sojourn time in the design of CR networks in smart grids.

Chapter 5

Denial-of-Service Attacks on Load

Frequency Control in Largely

Interconnected Power Grids

5.1 Introduction

As another reason that results in communication failures in smart grids, the impact

of the denial-of-service (DoS) attack on the load frequency control (LFC) in a largely

interconnected power grid is studied in this chapter. In particular, we consider DoS

attacks on the communication channels in the sensing loop through which measure-

ments telemetered in remote terminal units (RTUs) are sent to control centers. It can

be noted that adversaries may make the power system unstable by properly designing

their DoS attack sequences. Simulation studies are conducted to evaluate the effect

of DoS attacks on the dynamic performance of the power system.

The rest of this chapter is organized as follows. In Section 5.2, the DoS attack

issues in smart grids are introduced. In Section 5.3, the power system with DoS

attacks is modeled as a switched system. In Section 5.4, the existence of DoS attacks

that can make the power system unstable is proved. In Section 5.5, a two-area power

80

CHAPTER 5. DOS ATTACKS ON LFC 81

system model is used to evaluate the effect of DoS attacks on the power system.

Finally, conclusions are made in Section 5.6.

5.2 Denial-of-Service (DOS) Attacks in Smart

Grids

While open communication infrastructures are embedded into smart grids to support

vast amounts of data exchange, they make smart grids more vulnerable to cyber at-

tacks. The importance of securing current and future power grids has attracted more

and more attentions from both the academia and industry communities. In [72], the

authors pointed out that replacing proprietary network by open communication stan-

dards exposes process control and supervisory control and data acquisition (SCADA)

systems to cyber security risks. A class of false data attacks on state estimation

in power SCADA system, bypassing the bad data detection, were firstly present-

ed in [73]. In [74], adversaries were assumed to only know the perturbed model of

power systems when they are designing false data attacks against state estimations.

In [75], the smallest set of adversary-controlled meters was identified to perform an

unobservable attacks.

Although these works are very promising, they considered only static state esti-

mation in power systems without noticing the impact of attacks on the dynamics of

power systems. Regarding cyber attacks on SCADA control systems, a lot of chal-

lenges were identified by A. A. Cardenas et al in [74,76]. In [77], Y. Mo et al. studied

false data attacks on a control system equipped with a Kalman filter. As one of the

few automatic control systems in the control center of a power system, the LFC under

cyber attacks is considered in Viking projects conducted in [78, 79]. They performed

the analysis of the impacts of cyber attacks on control centers in a power system, by


Figure 5.1: Two-area load frequency control (LFC) under DoS attacks

using reachability methods. However, they only considered the scenario that control

center is attacked and controlled by adversaries. In fact, it is harder to attack the

control center than to compromise the communication channels in the sensing loop

of a power system. The sensing loop could be either for traditional SCADA systems

or wide-area measurement systems (WAMSs).

To launch a DoS attack on the communication channels, the adversaries can jam

the communication channels, attack networking protocols, and flood the network

traffic etc. [80, 81]. If attacked, measurement packets sent from sensors through this

channel will be lost. Thus, the existence of DoS attacks may destabilize power sys-

tems.

5.3 Modeling of A Largely Interconnected Power

System with DoS Attacks

In this section, the classical model of a LFC [8,33] is extended to include DoS attacks

existing in sensing channels of the multi-area interconnected power system, shown


in Fig 5.1. In Fig 5.1, the telemetered measurements for RTUs are sent back to the

control center of the LFC through communication channels either wired or wireless

networks. The adversaries can launch DoS attacks by jamming the communication

channels or flooding network traffics to cause congestions in networks. As the teleme-

tered measurements are lost, the control center cannot update its control commands

in time and the dynamic performance of the power system may be influenced. For

LFC studies, all the generators in each area are represented equivalently by one single

machine.

The model of the LFC in the area i was described in the previous chapter and

omitted here. For the whole multi-area power system, a linear time invariant(LTI)

interconnected model is given by:

x(t) = Acx(t) + Bcu(t) (5.1)

where

x =[

x1 x2 · · · xN

]T

; u =[

u1 u2 · · · uN

]T

;

Ac =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

A11 A12 · · · A1N

A21 A22 · · · A2N

... ... . . . ...

AN1 AN2 · · · AN

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

; Bc = diag

{B1 B2 · · · BN

}T

;

Due to the DoS attack and the attack sequence are inherently in a discrete-time

fashion, the following discrete-time model of the power system is considered:

x(k + 1) = Ax(k) + Bu(k) (5.2)

where, A = eAch, B =∫ h

0 eAcτ Bcdτ , h is the sampling period.


Figure 5.2: The model of the power system under DoS attacks

We consider the following optimal state feedback controller with the gain matrix

K

u(k) = −Kx(k) (5.3)

for the power system.

When DoS attacks are considered, the adversary attacks the communication chan-

nels, by preventing the sensed measurements in RTUs to be transmitted successfully

to the control center. We can model a DoS attack as a switching on/off event of the

state x(k) as shown in Fig 5.2. We denote the equivalent controller by

u(k) = −Kx(k). (5.4)

Since we consider the controller equipped with zero-order-hold (ZOH), the DoS

attack on x(k) can be modeled as follows.

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

x(k) = x(k) if, S1;

x(k) = x(k − 1) if, S2

(5.5)

Define the augmented state z(k) = [xT (k), xT (k − 1)]T . By integrating the (5.4)


into (5.2), we get the closed-loop form with the augmented state:

z(k + 1) = Φσiz(k) (5.6)

where σi is the switch position variable, σi = 1 for position S1, σi = 2 for position S2,

Φ1 =

⎡⎢⎢⎢⎢⎣

A − BK 0

I 0

⎤⎥⎥⎥⎥⎦; Φ2 =

⎡⎢⎢⎢⎢⎣

A −BK

0 I

⎤⎥⎥⎥⎥⎦.

In practice, DoS attacks can be performed by intentionally changing the switch

position for a random time interval [tsi, tfi), where tsi and tfi are the DoS attack

starting and finishing time instants, respectively. For example, in a time interval

[k, k + 30), the switch position may be in S1 during the time interval [k, k + 10], and

be in S2 during (k + 11, k + 30).

5.4 Existence of Successful DoS Attacks in the S-

mart Grid

In this section, it will be shown that DoS attacks can make the power system unsta-

ble by carefully designing the sequential attacking time intervals of DoS attacks. The

power system with DoS attacks has been modeled as a linear switched system in the

previous section. As it is known, the stability of a switched system has been exten-

sively addressed, such as [82,83]. On the one hand, the whole system which comprises

of several unstable subsystems can be stable by properly designing switching strategy

among these subsystems. On the other hand, it can be also unstable, by improperly

switching among several stable subsystems. From the adversaries’ point of view, they

may be able to make the whole power system unstable by choosing proper switching

strategies.


At first, we will show the necessary and sufficient conditions for the stability of a

linear switched system.

Theorem 5.4.1. [83] A linear switched system x(k + 1) = Φσix(k) where Φσi

∈{Φ1, Φ2, · · · , ΦN }, is asymptotically stable under arbitrary switching if and only if

there exists a finite integer n such that

||Φi1Φi2 · · · Φin|| < 1 (5.7)

for all n-tuple Φij ∈ {Φ1, Φ2, · · · , ΦN}, where j = 1, 2, . . . , n, {i1, i2, · · · , in} is the

switching rule.

According to the above Theorem 5.4.1, it may be possible for adversaries to find

switching rules to make the power system unstable as long as these switching rules

make ||Φi1Φi2 · · · Φin|| ≥ 1 happen. In fact, we can equivalently see the switched

system (5.6) as an average system

Φα = αΦ1 + (1 − α)Φ2 (5.8)

where 0 < α < 1.

Then, we can get the following theorem to show there might exist some switching

DoS attacks make the power system (5.6) unstable.

Theorem 5.4.2. The linear switched system (5.6) where Φσi∈ {Φ1, Φ2}, is unstable,

if there exists a constant 0 < α < 1 such that the average system Φα = αΦ1+(1−α)Φ2

has an eigenvalue with its magnitude outside the unity circle.

Proof. The time interval [ts, tf), defining η = tf − ts, is considered for the linear

switched system (5.6) where Φσi∈ {Φ1, Φ2}. We assume, without attacks, the linear

switched system (5.6) stays at Φ1 for αη seconds. Then, the adversary starts the DoS


attacks. That means the system (5.6) stays at Φ2 for (1 − α)η seconds. The state of

the system (5.6) at time instant tf will be

x(tf ) = eΦ1αηeΦ2(1−α)ηx(ts) (5.9)

Let Φ(tf ) = eΦ1αηeΦ2(1−α)η . The system (5.6) is unstable, if Φ(tf ) has eigenvalues

which are outside the unity circle. For some commutable and Hurwitz matrices Φ1

and Φ2,

Φ(tf ) = eΦ1αηeΦ2(1−α)η = eΦ1αη+Φ2(1−α)η = e(Φ1α+Φ2(1−α))η (5.10)

Thus, the system (5.6) is unstable, if its equivalent average system matrix Φ1α +

Φ2(1 − α) has eigenvalues with magnitudes outside the unity circle.

5.5 Simulations

In this section, a two-area power system model shown in Fig 5.1 is used to evaluate the

impacts of DoS attacks on power systems. The generators in each area are modeled as

a single equivalent generator. Matlab/Simulink R2012a is chosen as the simulation

environment. In this chapter, we consider DoS attacks existing in communication

channels of the sensing loop of the power system. We use 100 MVA as the base unit

for per unit (pu) calculations. All the parameters of the two-area power system are

shown as follows:

Tch1,2 = 0.08s, Tg1,2 = 0.3s, R1,2 = 2.4Hz/pu, D1,2 = 0.0084pu/Hz

M1,2 = 0.1667, T12 = 0.08678.

The original linear quadratic optimal controller is u = −Kx. By using the

command dlqr in Matlab R2012a and setting Q = 100 ∗ diag([1111111111]),R =

100 ∗ diag([11]), the controller gain K can be obtained.

Firstly, according to Theorem 5.4.2 and its proof, the root locus of det(λI−Φα) = 0


−0.5 0 0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

Real(1/s)

Imag

(rad

/s) α=0.4

α=0.4

Figure 5.3: The root locus of the average two-area power system (Arrows indicate αdecreasing)

, where Φα = αΦ1 + (1 − α)Φ2, is shown in Fig 5.3 by increasing α ∈ [0.1, 1). As it is

illustrated in Fig 5.3, the system becomes unstable as α ≤ 0.4.

Then, simulations are conducted to evaluate the system performance when both

area 1 and area 2 are under DoS attacks. We will see how the dynamics change

according to different DoS attacks launching time, indexing by 0 < α ≤ 1. The time

duration is set to be 1 second. Four cases are considered in this chapter:

• Case 1: α = 1, the power system operates normally;

• Case 2: α = 0.8, the power system operates under DoS attacks activated at

T = 0.8s;


T = 0.4s;


T = 0.3s.

The dynamics of the two-area power system are shown in Fig 5.4 and Fig 5.5. These


0 0.5 1−0.2

−0.15

−0.1

−0.05

0

Time (Ts=0.01s)

Δ f 1 (p

u)0 0.5 1

0

0.5

1

Time (Ts=0.01s)

Δ P m

1 (pu)

0 0.5 10

0.5

1

1.5

Time (Ts=0.01s)

Δ P v1

(pu)

0 0.5 10

0.2

0.4

Time (Ts=0.01s)

Δ P tie1

(pu)

α=1 & 0.8

α=0.4α=0.3

α=0.3 α=0.4

α=0.3

α=0.4

α=1 & 0.8

α=1 & 0.8

α=1 & 0.8

α=0.3 α=0.4

Figure 5.4: The dynamics of area 1 under different DoS attacks initial times

figures illustrate the effects of the proposed DoS attacks on the two-area dynamic

power system. It can be observed that DoS attacks affect the dynamics of the power

system seriously when α ≤ 0.4s which coincide with the root locus analysis. Thus,

the adversary would like to launch DoS attacks as early as possible. The DoS attacks

might not work anymore if they start late such as the cases when α = 0.8s in these

two figures.

5.6 Summary

This chapter considers the problem that how DoS attacks in the cyber layer of smart

grids can affect the dynamic performance of physical power systems. The power

system under DoS attacks is modeled as a linear switched system, by formulating DoS

attacks as a switch (on/off) action on sensing channels. We identify the existence of


0 0.5 1−0.2

−0.15

−0.1

−0.05

0

Time (Ts=0.01s)

Δ f 2 (p

u)0 0.5 1

0

0.5

1

Time (Ts=0.01s)

Δ P m

2 (pu)

0 0.5 10

0.5

1

1.5

Time (Ts=0.01s)

Δ P v2

(pu)

0 0.5 10

0.2

0.4

Time (Ts=0.01s)

Δ P tie2

(pu)

α=0.4

α=0.4

α=0.4

α=1 & 0.8α=1 & 0.8

α=1 & 0.8α=1 & 0.8

α=0.3

α=0.3

α=0.4

α=0.3

α=0.3

Figure 5.5: The dynamics of area 2 under different DoS attacks initial times

DoS attacks that can destabilize the power system, by using switched system theories.

In simulation studies, a two-area LFC power system model has been built to evaluate

the effect of DoS attacks with different attack-launching instants. It shows that DoS

attacks can affect the dynamic performance of the power system badly if they are

launched early before the dynamics of the power system converge. Its dynamics are

weakly influenced when adversaries enable DoS attacks after the dynamics of the

power system converge.

Chapter 6

Modeling and Distributed Gain

Scheduling Strategy for Load Frequency

Control with Communication Failures in

Largely Interconnected Power Grids

6.1 Introduction

From analysis results presented in the previous two chapters, it can be found that

it is critical to design advanced control methods to improve the robustness of smart

grid control to communication failures. In this chapter, a distributed gain schedul-

ing LFC strategy is proposed to compensate for the degraded performance due to

communication failures. A four-area LFC power system model is used to verify the

effectiveness of the proposed method.

The rest of this chapter is organized as follows. In Section 6.2, the general structure

of the proposed distributed gain scheduling approach is introduced. In Section 6.3,

a new power system model is introduced to integrate the changes of communication

topologies into the physical dynamics. In Section 6.4, the stability analysis of this new

power system is conducted. The distributed gain scheduling algorithm is described

91

CHAPTER 6. DISTRIBUTED GAIN SCHEDULING FOR LFC 92

in Section 6.5. In Section 6.6, simulations are conducted by using a four-area power

system model under six communication topologies. Finally, Section 6.7 concludes this

chapter with some remarks.

Figure 6.1: Centralized control scheme

Figure 6.2: Distributed control scheme


Figure 6.3: The proposed distributed control algorithm

6.2 General Structure of The Proposed Distribut-

ed Gain Scheduling Strategy

To improve the monitoring/operation and the robustness of a smart grid to random

faults, it is well agreed that distributed control approaches (shown in Fig 6.2) work

better than their centralized counterparts (see Fig 6.1) [2, 48, 84]. For distributed

control strategies, the entire power system can be divided into multiple interconnected

areas [28, 35, 85] or micro-grids with distributed generations (DGs) [2, 86, 87]. Each

area is a subsystem with its own control center. Two-way communications of sensing

measurements and control inputs are needed within each area and among different

areas. To support the vast amounts of information exchanged in a real-time power

system, high-speed open communication infrastructures are urgently required to be

implemented in large-scale power systems [51, 88]. In [85], GridStat, a prototype of

a new communication framework, is proposed for delivering real-time information

and operational commands in power systems. For substation automation, local area


networks (LANs) are introduced in communications for intelligent electronic devices

(IEDs) within substations under the communication standard IEC 61850, while wide

area networks (WANs) are used for data exchange among substations [54, 89, 90].

In this chapter, a new distributed control method is proposed for the LFC of a

smart grid to counteract the communication failure effect. A two-layer structure is

introduced and a global communication topology detector (CTD) is used to check

communication topologies in a real-time fashion and distribute the communication

topology information to each area control center (ACC). With the knowledge of its

local communication topology changes, each ACC calculates its optimal local and

inter-connected feedback gain matrices accordingly. The proposed distributed control

scheme is illustrated in Fig 6.3.

6.3 Modeling of A Multi-area Interconnected

Power System with Communication Failures

The frequency control of a power system is fundamental in determining the way in

which the frequency will change when load changes happen [91, 92].

We assume there are N interconnected areas in the power system. We write the

discrete-time state-space model of the above dynamics for the LFC in area i as follows:

xi(k + 1) = Aixi(k) + Biui(k) +N∑

j=1,j �=i

Aijxj(k) + FiΔPLi(k), i ∈ {1, 2, · · · , N} (6.1)

where xi =[

Δfi ΔPmiΔPvi

ΔP ijtie

]T

; ui = ΔPci;


Ai =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

− Di

Mi

1Mi

0 − 1Mi

0 − 1Tchi

1Tchi

0

− 1RiTgi

0 − 1Tgi

0

− N∑j=1,j �=i

Tij 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

; Aij =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0

0 0 0 0

0 0 0 0

Tij 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

;

Bi =[

0 0 1Tgi

0

]T

; Fi =[

− 1Mi

0 0 0

]T

.

In the distributed control of the power system, each area controller needs not only

its own states, but also the states of its neighbors. By assuming the communication

infrastructures are completely reliable, the distributed controller for ith area can be:

ui(k) = −Kixi(k) −N∑

j=1,j �=i

Kijxj(k), i ∈ {1, 2, · · · , N} (6.2)

where Ki, Kij are constant feedback gains matrices.

For ui in the ith ACC, other areas’ states xj are accessible through the commu-

nication infrastructures. However, random communication failures happen in com-

munication links. They will cause communication topology changes. Therefore, it

is more practical to take communication topology changes into consideration when

we design the distributed control for the LFC in a smart grid. The structure dis-

turbances of ui resulted from communication topology changes are modeled by the

following time-varying communication matrix.

Definition. 6.3.1 Communication matrix: The communication matrix of S is a

binary matrix L(k) = [lij(k)]n×n have the elements lij(k) = 0 or 1. Since xi is a local

state in the ith ACC, lii(k) = 1 is always true. That means L(k)’s diagonal elements


are all ones.

L(k) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 l12(k) · · · l1n(k)

l21(k) 1 · · · l2n(k)

... ... . . . ...

ln1(k) ln2(k) · · · 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(6.3)

Now the dynamics of the large power system including communication topology

effects are written as

xi(k + 1) = Aixi(k) + Biui(k) +N∑

j=1,j �=i

lij(k)Aijxj(k) + FiΔPLi(k), i ∈ {1, 2, · · · , N}

(6.4)

Also, the state feedback distributed controllers including the effects of communication

topologies are

ui(k) = −Kixi(k) −N∑

j=1,j �=i

lij(k)Kijxj(k), i ∈ {1, 2, · · · , N} (6.5)

where Ki, Kij are constant feedback gains matrices. Plugging these controllers into

the dynamics of the large power system, we get the closed-loop form as follows.

xi(k + 1) = Aixi(k) +N∑

j=1,j �=i

lij(k)Aijxij(k) + FiΔPLi(k), i ∈ {1, 2, · · · , N} (6.6)

where Ai = Ai − BiKi,Aij = Aij − BiKij.


6.4 Stability Analysis of The Multi-area Inter-

connected Power System with Communication

Failures

In this section, we give sufficient conditions to keep the multi-area interconnected

power system with communication topology changes (its dynamic model is (6.6))

globally asymptotically stable.

In order to analyze the stability of the power system (6.6), we firstly recall the

theorem of M − matrix.

Lemma 6.4.1. [93] There exists a positive diagonal matrix D such that DW +W T D

is positive definite if and only if W is an M − matrix; that is the leading principal

minors of W are positive:

det

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

w11 w12 · · · w1N

w21 w22 · · · w2N

... ... . . . ...

wN2 w12 · · · wNN

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

> 0,

In [94] and [93], both Siljak and Khalil analyzed the stability issues of nonlin-

ear interconnected systems by a Lyanunov function method. A similar Lyanpunov

method is also used for the stability analysis of the multi-area interconnected power

system (6.6).


Theorem 6.4.2. A linear system (6.6) is globally asymptotically stable if the follow-

ing test matrix W = [wij]N×N is an M − matrix.

wij =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

λm(Hi − Gi) − 2lijλM(Hi)λ1/2M (AT

i Ai), i = j

−2lijλM(Hi)λ1/2M (AT

ijAij), i = j

(6.7)

where Gi,Hi are symmetric positive definite matrices satisfy the Lyapunov matrix

equations AiHiAi −Hi = −Gi, and λm(�) and λM(�) are the minimum and maximum

eigenvalues of the corresponding matrices respectively.

Proof. The linear system (6.6) is an interconnection of N subsystems

xi(k + 1) = Aixi(k), i ∈ {1, 2, · · · , N} (6.8)

Denote interconnection terms as functions Ii(x(k)) =N∑

j=1,j �=ilijAijxj(k),

where x = [xT1 , xT

2 , · · · , xTN ]T .

Let us build the following function

V (x(k)) =N∑

i=1diVi(xi(k)) (6.9)

as a composite Lyapunov function of the linear system (6.6), where di > 0 for all

i ∈ {1, 2, · · · , N}.

For each subsystem i ∈ {1, 2, · · · , N}, we choose a quadratic Lyapunov function

Vi(xi(k)) = xi(k)T Hixi(k) (6.10)

where Hi is a symmetric positive definite matrix. Hi satisfies a Lyapunov matrix

equation AiHiAi − Hi = −Gi, where Gi is a symmetric positive definite matrix.


Each subsystem is stable, if the following inequalities hold.

λm(Hi)||xi(k)||2 ≤ Vi(xi(k)) ≤ λM(Hi)||xi(k)||2, i ∈ {1, 2, · · · , N} (6.11)

where, λm(Hi) and λM(Hi) are the minimum and maximum eigenvalues of Hi respec-

tively. And,

ΔVi(xi(k)) = Vi(xi(k + 1)) − Vi(xi(k)) ≤ −λm(Gi)||xi(k)||2, i ∈ {1, 2, · · · , N} (6.12)

where, λm(Gi) is the minimum eigenvalue of Gi respectively.

Define a comparison function φi ∈ K for each i ∈ {1, 2, · · · , N}. We suppose the

interconnection terms Ii(x(k)) =N∑

j=1,j �=ilijAijxj(k) satisfy the following inequalities

||Ii(x(k))|| ≤N∑

j=1,j �=i

lijξijφj(||xj(k)||), i ∈ {1, 2, · · · , N} (6.13)

where ξij ≥ 0. The derivative of V (x(k)) =N∑

i=1diVi(xi(k)) along the trajectories of

the multi-area interconnected power system (6.6) satisfies the inequality

ΔV (x(k)) = V (x(k + 1)) − V (x(k))

≤ N∑i=1

di [−ηiφ2i (||xi(k)||) + lijξijφi(||xi(k)||)φj(||xj(k)||)]

≤ −1/2φT (||x(k)||)(DW + W T D)φ(||x(k)||)

(6.14)

where φ(||x(k)||) = [φ1(||x1(k)||), φ2(||x2(k)||), · · · , φN(||xN(k)||)]T , D =

diag{d1, d2, · · · , dN}, ηi > 0. Define ξij = λ1/2M (AT

ijAij). From the inequalities

(6.11) and (6.12), we choose φi(xi(k)) = ||xi(k)||. Thus, we get a W matrix whose


elements are defined by

wij =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

λm(Hi − Gi) − 2lijλM(Hi)λ1/2M (AT

i Ai), i = j

−2lijλM(Hi)λ1/2M (Aij

TAij), i = j

(6.15)

If W is M −matrix, according to the Theorem 6.4.2, the inequality DW +W T D > 0

holds. Then, ΔV (x(k)) < 0 holds for all x = 0 . So, the interconnected system is

uniformly asymptotically stable.

6.5 Distributed Gain Scheduling Strategy for The

LFC in A Smart Grid

In this chapter, our goal is to develop a distributed gain scheduling method for the

LFC of a smart grid to compensate for the degraded dynamic performances caused

by communication topology changes. As shown in Fig 6.3, a global communica-

tion topology detector (CTD) is used by installing some network topology softwares

(Remark 6.5.1 ). Then, it distributes the communication topology to each ACC by

sending the ith row vector information Ri of the communication matrix to the ith

ACC. After knowing its local neighbors, the ith ACC calculates corresponding opti-

mal local and inter-connective gain matrices Ki(∞) and K∗ij to reduce the effects of

the corresponding inter-connective terms.

Firstly, we assume that each pair (Ai, Bi) as defined in equation (6.1) is control-

lable. Thus, we can always choose the feedback gains Ki to place the eigenvalues

of Ai at the desired locations in root loci. Linear quadratic regulators (LQR) are

designed for each (Ai, Bi). That is to find optimal state feedback controllers ui(k) for

all i ∈ {1, 2, · · · , N} to minimize the following cost functions during the time range


[0, tf ]:

Ji = xi(tf)T Sixi(tf ) +tf −1∑k=0

{xi(k)T Qixi(k) + ui(k)T Riui(k)}, i ∈ {1, 2, · · · , N} (6.16)

where Si, Qi are semi-definite symmetric matrices, and Ri are positive definite sym-

metric matrix.

Based on the finite time horizon cost functions, LQR state feedback actions u(k)

are given by

ui(k) = −Ki(k)xi(k), i ∈ {1, 2, · · · , N} (6.17)

where

Ki(k) = (Ri + BTi Pi(k)Bi)−1BT

i Pi(k)Ai, i ∈ {1, 2, · · · , N} (6.18)

We solve the following algebraic Riccati equation (ARE) to get the positive definite

matrices Pi(k):

Pi(k) = Qi +ATi (Pi(k+1)−Pi(k+1)Bi(Ri +BT

i Pi(k+1)Bi)−1BTi Pi(k+1))Ai (6.19)

with terminal conditions Pi(tf ) = Si.

In order to guarantee the real-time performance of the power system and to im-

prove the convergence rate of state variables, we choose the steady state feedback

gain Ki(∞), instead of feedback gain sequences Ki(t).

Ki(∞) = (Ri + BTi Pi(∞)Bi)−1BT

i Pi(∞)Ai, i ∈ {1, 2, · · · , N} (6.20)

We solve the following algebraic Riccati equation (ARE) to get the positive definite


matrices Pi(∞):

Pi(∞) = Qi+ATi (Pi(∞)−Pi(∞)Bi(Ri+BT

i Pi(∞)Bi)−1BTi Pi(∞))Ai, i ∈ {1, 2, · · · , N}

(6.21)

Because Pi(∞) and Ki(∞) are constant in the steady state, there is no need to

compute the feedback gains recursively. It reduces the computation load without

influencing convergence performances.

To conduct the stability analysis for the power system, we need to find a test

matrix W to check whether it is an M-matrix, according to the Theorem 6.4.2. In

order to get a simple test matrix W , We transform Ai into diagonal form Λi by a

linear nonsingular transformation

xi(k) = Tixi(k), i ∈ {1, 2, · · · , N} (6.22)

where, Ti = [ti1, ti2, · · · , tin], tik is the kth right eigenvector of Ai corresponding to

the kth eigenvalue λik = −σik ± jωi

k or λik = −σik, k ∈ {1, 2, · · · , n}

The diagonal formed system is written as

xi(k + 1) = Λixi(k) +N∑

j=1,j �=i

Δij xj(k), i ∈ {1, 2, · · · , N} (6.23)

where Λi = T −1i AiTi, Δij = T −1

i AijTj .

We define a quadratic Lyapunov function as follows,

Vi(xi(k)) = xi(k)T Hixi(k), i ∈ {1, 2, · · · , N} (6.24)

where Hi is the solution of the Lyapunov matrix equation

ΛTi HiΛi = Hi − Gi (6.25)


where, let Hi = I, Gi = diag{1 − σi1, · · · , 1 − σi

n}.

Then, the test matrix W = [wij]N×N is defined as

wij =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

σiM − 2lijλ

1/2M (ΔT

iiΔii), i = j

−2lijλ1/2M (ΔT

ijΔij), i = j

(6.26)

The diagonal formed system is stable if W is an M-matrix. The original system is

stable if the diagonal formed system is stable.

Also, we select the optimal K∗ij when the element of communication matrix lij = 1

by using the diagonal formed system.

To perform the distributed gain scheduling algorithm for each area controller ACC,

it only needs to know the ith row of the communication matrix. Thus, it becomes

simpler than solving it in the centralized method.

When lij = 1, we choose the optimal K∗ij by solving the following optimization

problem.

ζ∗ij = min{||Δij − BiKij||} (6.27)

where Δij = T −1i AijTj,Bi = T −1

i Bi,Kij = KijTj.

After calculating each optimal K∗ij , we get the optimal K∗

ij by

K∗ij = K∗

ijT−1j (6.28)

Assuming the initial communication topology is known to the CTD, the dynamic

gain scheduling algorithm in each ACC is summarized as in Algorithm 1.


Algorithm 1 Dynamic gain scheduling procedure1: For each given time period b

2: The global CTD checks the current communication topology and forms it in the

communication matrix formula L = [RT1 , RT

2 , · · · , RTn ]T .

3: It distributes ith row vector RTi to the corresponding local ACC.

4: Each ACC receives its row vector and calculates its system matrices (AT ii , BT i

i )

, where Ti denotes the ith communication topology.

5: Each ACC calculates its local LQR gain Ki(∞) and the optimal inter-connective

gains K∗ij according to the minimum optimization formula for each lij = 1.

6: end .

Remark 6.5.1 In practice, a variety of software products already exist to au-

tomatically detect the real-time communication topology using Virtual Routers and

Routing Protocol Listening techniques, such as HP Network Node Manager i, OpMan-

ager, and OPNET’s NetMapper etc.. By implementing these software applications,

real-time communication topology information can be obtained by the CTD and sent

the information to every ACC in each area in smart grids, as shown in Fig 6.3.

6.6 Simulations

In this section, a four-area power system model (see Fig 6.3) is used to evaluate

the proposed control method. The mean square error (MSE) of the state vector

xi =[

Δfi ΔPmiΔPvi

ΔP ijtie

]T

in each area i ∈ {1, 2, 3, 4} is chosen as the

dynamic performance metric for the four-area power system. The generators in each

area are modeled as a single equivalent generator. The simulations are run in the

Matlab R2012a environment. All the parameters of the four-area power system are

given in Appendix D. In this study, 100 MVA is the base unit for per unit (pu)

calculations. All the power system models are sampled by sampling period T = 0.01s.


Each simulation lasts for 12s. The initial frequency deviations for all the four area

are 0.5Hz.

Firstly, the impacts of communication failures on the dynamic performances of

the four-area power system are analyzed. 6 communication topologies (see Fig 6.4)

are chosen. Fig 6.4(a) is the ideal communication topology of the fully connected

distributed controller for the power system. The dynamic performance of this com-

munication topology is used as the reference when we calculate the MSEs of the state

vector xi in each area i ∈ {1, 2, 3, 4}. Fig 6.4(b-f) are 5 imperfect communication

topologies which are caused by communication failures among the four areas in the

power system. The five imperfect communication topologies are triggered by the fol-

lowing scheduling illustrated in Fig 6.5. In Fig 6.5, mode 1-5 represent the imperfect

communication topologies shown in Fig 6.4(b-f), correspondingly. With comparison

to the fully connected communication topology (see Fig 6.4(a)), the MSEs of the

state vector xi in area i ∈ {1, 2, 3, 4} under the 5 imperfect communication topologies

(see Fig 6.4(b-f)) are shown from Fig 6.6 to Fig 6.9. As illustrated in these figures,

the distributed controllers for the four-area power are sensitive to communication

topology changes. Also, as indicted in these results, the dynamic performance of the

distributed controller for each area greatly depends on how well it can communicate

with its neighbor areas. For instance, as it can be seen from Fig 6.7, the maximal

and sub-maximal peak mean square errors of the state vector x2 appear when it are

under the topologies of Fig 6.4(e) and Fig 6.4(f). The reason for these results is the

distributed controller in area 2 under the topologies of Fig 6.4(e) and Fig 6.4(f) lacks

the inter-area state information.

To improve the robustness of the distributed controller to communication topol-

ogy changes, we design then the distributed gain scheduling strategy as described in

previous sections. The five imperfect communication topologies (see Fig 6.4(b-f)) are

triggered by the same schedule as shown above (see Fig 6.5). The MSEs of the state


Figure 6.4: Six communication topologies

0 2 4 6 8 10 120

1

2

3

4

5

6

Time (Ts=0.01s)

Com

mun

icat

ion

topo

logy

mod

e

Figure 6.5: The scheduling scheme of the 5 imperfect communication topology modes


0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Time (Ts=0.01s)

Mea

n sq

uare

err

ors

under topology Fig 6.4(b)under topology Fig 6.4(c)under topology Fig 6.4(d)under topology Fig 6.4(e)under topology Fig 6.4(f)

Figure 6.6: Dynamic response of Area 1

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Time (Ts=0.01s)

Mea

n sq

uare

err

ors




0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

Time (Ts=0.01s)

Mea

n sq

uare

err

ors



0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Time (Ts=0.01s)

Mea

n sq

uare

err

ors




0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time (Ts=0.01s)

Mea

n sq

uare

err

ors

Area1

Area2

Area3

Area4

Figure 6.10: Dynamic responses of 4 areas under the distributed gain schedulingstrategy

vector xi in area i ∈ {1, 2, 3, 4} are shown in Fig 6.10. As it can be noticed from

Fig 6.10, the MSEs in the four areas are greatly reduced. These results illustrate the

proposed strategy can enhances the robustness of the LFC.

6.7 Summary

In this chapter, the modeling and distributed control problems for a smart grid with

communication failures are addressed. The change of communication topologies in

the smart grid is modeled as a time-varying communication topology matrix. This

communication topology matrix enables to build a closed-loop power system model,

integrating the dynamic communication topology into the dynamics of physical pow-

er systems. The stability analysis of the closed-loop power system is conducted. A

distributed gain scheduling strategy for the LFC is proposed to compensate for the


degraded performances of the smart grid. Simulation results of a four-area power sys-

tem under six communication topologies have confirmed that the proposed approach

is able to maintain low mean-square errors for the power system with communication

failures.

Chapter 7

Conclusions

In this chapter, we draw conclusions and discuss several possible topics for the future

research.

7.1 Thesis Conclusions

• 1. The effect of communication delays on the LFC in an islanded multi-DG

microgrid is studied. As communication delays are taken into account, a small-

signal model based method is applied to determine a delay margin below which

the microgrid can maintain stable. By performing a thorough theoretical analy-

sis, relationships between secondary frequency control gains and delay margins

are identified. Simulations results of this multi-DG microgrid system have ver-

ified that communication delays can badly affect the stability of the microgrid

and illustrated the correctness of the proposed small-signal analysis results for

the microgrid.

• 2. Based on the small-signal model of the microgrid formulated in the previous

chapter, a gain scheduling approach is proposed to compensate the communi-

cation delay effect on the LFC performance of the microgrid. This approach

consists of both offline stability analysis and online gain schedule. In the offline

111

CHAPTER 7. CONCLUSIONS 112

stability analysis, feasible gain sets corresponding to different communication

delays are found based on the root locus of the small-signal model of the micro-

grid. Relationships between these gains and the degraded performance indexed

by the average quadratic state error are also identified by series of simulation-

s. In the online gain schedule, via the GPS service supplied by PMUs and

gain schedulers, each local controller can calculate current time delays and then

schedule the corresponding controller gains corresponding to the current time

delay. Studies of an islanded microgrid with 4 inverter-based DGs have shown

the proposed gain scheduling method can greatly improve the dynamic perfor-

mance of the microgrid, with comparison to a fixed gain secondary frequency

controller.

• 3. For largely interconnected power systems, a CR network is considered as the

source of communication failures. By modeling the CR network as a On-Off

switch with sojourn times, a novel switched power system model is proposed for

the LFC of an interconnected power system. To study the stability of the power

system, two main types of CR networks are considered, including the sojourn

times that are arbitrary but bounded and that follow independent and identical

distribution (i.i.d). The sufficient conditions are obtained for the stability of the

power system with these two kinds of CR networks, respectively. Simulation

studies show that the obtained results are very useful to the design of CR

networks in order to guarantee the stochastic stability of the power systems.

• 4. The DoS attack is considered as another reason that results in communication

failures for the largely interconnected power systems. The state-space model of

a power system under DoS attacks is formulated as a switched system. Based

on switched system theories, the existence of DoS attacks that may cause the

dynamics of a power system unstable is proved. A two-area power system is


used to conduct case studies.

• 5. A distributed gain scheduling strategy is proposed to compensate the po-

tential degradation of the performance of the LFC caused by communication

failures in largely interconnected power systems. This strategy has a two-layer

structure. In its cyber layer, a global communication topology detector (CTD)

is used to check communication topologies in a real-time fashion and distribute

the communication topology information to each area control center (ACC) in

the power systems. With the knowledge of its local communication topology

changes, in the physical layer of this structure, each ACC calculates its optimal

local and inter-connected feedback gain matrices accordingly. A four-area smart

grid model is used to verify the effectiveness of the proposed method. Simu-

lation results show that the proposed distributed gain scheduling approach is

capable to improve the robustness of the smart grid to communication topology

changes caused by communication failures.

7.2 Possible Directions for Future Research

The following are possible topics for the future research.

• 1. The effects of communication delays on the performance of an islanded

microgrid are analyzed in Chapter 2. In fact, the communication delays result

from complicated networking actions in a communication system. Therefore, it

would be interesting to analyze the effects of the factors of the networking layer

of the communication system on the microgrid performance, including TCP/IP

protocols and UDP protocols.

• 2. The stability of a largely interconnected power system under denial of service

(DoS) attacks has been investigated in Chapter 5. It would be very interesting to


investigate some networking defense mechanisms to act against the DoS attacker

in the communication system through which measurements and control signals

of the power system are transmitted. When we design these defense mechanisms

in the communication system, the dynamic performance of the physical power

system should also be taken into account.

• 3. In this thesis, the approaches for the compensations of communication delays

and failures in smart grids are considered from the power system control aspect.

In specific, these approaches are gain scheduling control algorithms. It would

be interesting to consider reducing communication delays and failures from the

aspect of the communication system design, including improving the network-

ing protocols in the network layer of the communication system and queueing

mechanisms in the physical layer of the communication system.

Appendix A

Microgrid Parameters

Table A.1: Distribution system parameters

Parameters Value

Sbase 10 (MVA)

Vbase1 120√

2/√

3 (kV)

Vbase2 12.5√

2/√

3 (kV)

Vbase3 208√

2/√

3 (V)

RS 0 (p.u.)

XS 0 (p.u.)

RT 0 (p.u.)

XT 0.1 (p.u.)

Rf 0.0029 (p.u.)

Xf 0.0041 (p.u.)

Rt 0 (p.u.)

Xt 0.2 (p.u.)

Table A.2: Inverter parameters

Parameters Value

KpP LLi 50

KiP LLi 500

Kpii 2.5

Kiii 500

LS 1 (mH)

Kppi 2.5

Kipi 100

ω0 377 (rad/s)

Qref1, Qref2, Qref3, Qref4 0 (p.u.)

115

Appendix B

Chebyshev’s Differentiation Matrix

A N + 1 × N + 1 dimension Chebyshev’s differentiation matrix DN is created as

follows, according to [17].

Firstly, N + 1 Chebyshev’s nodes have to be defined by normalizing the interpo-

lation points on the interval [−1, 1]:

xk = cos(kπ

N), k = 0, · · · , N. (B.1)

Then, the element (i, j) in the differentiation matrix DN indexed from 0 to N is

defined as the following:

D(i,j) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ci(−1)i+j

cj(xi−xj) , i = j

−12

xi

1−x2i, i = j = 1, N − 1

2N2+16 , i = j = 0

−2N2+16 , i = j = N

(B.2)

where c0 = cN = 2 and c2 = · · · = cN−1 = 1. In Chapter 3 of this thesis, D2 is used

116

APPENDIX B. CHEBYSHEV’S DIFFERENTIATION MATRIX 117

for calculations. It is

D2 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

32 −2 1

2

12 0 −1

2

−12 2 −3

2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (B.3)

Appendix C

Two-area Power System Parameters

The two-area power system parameters are shown as follows.

Table C.1: Two-area power system parameters

Area 1 Area 2

Tch1 = 0.4s Tch2 = 0.36s

Tg1 = 0.08s Tg2 = 0.30s

R1 = 2.4Hz/pu R2 = 2.4Hz/pu

D1 = 0.014pu/Hz D2 = 0.0084pu/Hz

2H1 = 0.2pu · s 2H2 = 0.1667pu · s

T12 = 0.08678 T21 = 0.08678

118

Appendix D

Four-area power system parameters

The four-area power system parameters are shown as follows.

Table D.1: Four-area power system parameters

Area 1,3 Area 2,4

Tch1,3 = 0.17s Tch2,4 = 0.2s

Tg1,3 = 0.4s Tg2,4 = 0.35s

R1,3 = 0.0 R2,4 = 0.05

D1,3 = 1.5 D2,4 = 1.8

M1,3 = 12 M2,4 = 12

B1,3 = 41.5 B2,4 = 61.8

Tij = 0.05 Tij = 0.05

119

Appendix E

Publications

The research presented in this thesis resulted in a number of refereed publications,

both in journals and conference proceedings.

1. Shichao Liu, Peter X. Liu, and A. El Saddik, “Modeling and distributed gain

scheduling strategy for load frequency control in smart grids with communica-

tion topology changes", ISA Transactions, vol. 53, no. 2, pp. 454-461, 2014

2. Shichao Liu, Peter X. Liu, and A. El Saddik, “A stochastic game approach to

the security issue of networked control systems under jamming attacks", Journal

of The Franklin Institute, Accepted, In Press, 2014

3. Shichao Liu, Peter X. Liu, and A. El Saddik, “Modeling and stability analysis

of automatic generation control over cognitive radio networks in smart grids",

IEEE Transactions on Systems, Man, and Cybernetics: Systems, Accepted, In

Press, 2014

4. Shichao Liu, Xiaoyu Wang, and Peter X. Liu, “Impact of communication delays

on secondary frequency control in an islanded microgrid", IEEE Transactions

on Industrial Electronics, under Minor Revision

120

APPENDIX E. PUBLICATIONS 121

5. Shichao Liu, Peter X. Liu, and A. El Saddik., “Modeling and stochastic control

of networked control system with packet losses," 2011 IEEE Instrumentation

and Measurement Technology Conference (I2MTC), China, 10-12 May 2011,

pp. 1-5

6. Shichao Liu, Peter X. Liu, and A. El Saddik, “Modeling and dynamic gain

scheduling for networked systems with bounded packet losses ", 2011 IEEE

International Workshop on Measurements and Networking Proceedings, Italy,

2011, pp. 135-139.

7. Shichao Liu, Peter X. Liu, and A. El Saddik, “Load frequency control for wide

area monitoring and control system (WAMC) in power system with open com-

munication links," 2012 IEEE Power Engineering and Automation Conference

(PEAM), China, 18-20 Sept. 2012, pp. 1-5

8. Shichao Liu, Peter X. Liu, and A. El Saddik, “Denial-of-Service (DoS) attacks

on load frequency control in smart grids ", 2013 IEEE PES Innovative Smart

Grid Technologies (ISGT), Washington D.C, America, 2013, pp. 1-6.

9. Shichao Liu, Peter X. Liu, and A. El Saddik, “A stochastic security game for

Kalman filtering in networked control systems (NCSs) under Denial of Service

(DoS) attacks (invited paper)", 2013 IFAC International Conference on Intel-

ligent Control and Automation Science (ICONS 2013), Chengdu, China, 2013,

pp. 1-6

10. Xinran Zhang, Shichao Liu, Wes Kwasnicki, Yu Cui, Xiaoyu Wang, Chao Lu,

“Wide-area HVDC damping controller design in Alberta power grid", 2014

CIGRÉ Canada Conference, Toronto, Canada, September, 2014, to be pre-

sented

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